1. Group actions and other topics in group theory

October 11, 2014

The main topics considered here are group actions, the Sylow theorems, semi-directproducts, nilpotent and solvable groups, and simple groups.

See Preliminary remarks for some of the notation used here, especially regarding generallinear groups. Some further notation: [n] denotes the set of the first n natural numbers1, 2, ..., n.Pk[n] denotes the set of k-element subsets of [n].

1 Group actions

1.1 Definition of a group action or G-set

Let G be a group, with identity element e. A left G-set is a set X equipped with a mapθ : G×X−→X satisfying (i) θ(gh, x) = θ(g, θ(h, x)) for all g, h ∈ G and all x ∈ X, and (ii)θ(e, x) = x for all x ∈ X. Usually we write either g ·x or simply juxtaposition gx for θ(g, x);in the latter notation conditions (i) and (ii) become (gh)x = g(hx) and ex = x. We also callthis data a group action, or say that “G acts on X” (on the left).

Similarly a right G-set is a set X equipped with a map θ : X × G−→X satisfying (inthe evident juxtaposition notation) x(gh) = (xg)h and xe = x. Of course the distinctionbetween left and right G-actions does not depend on whether we write the domain of θ asG×X or X ×G. The distinction is that in a left action gh acts by h first, then g, whereasin a right action g acts first, then h.

Notice that up to this point, we haven’t even used the existence of inverses, so exactlythe same definition makes sense for left and right monoid actions. We will make little useof monoid actions, however. One immediate advantage of the existence of inverses is thatany right action can be converted to a left action by setting g · x = xg−1; similarly any leftaction can be converted to a right action. Nevertheless it is important to pay close attentionto which side the group is acting on. If the side is not specified we always mean a left action(an arbitrary choice on my part!). Any statement about left actions has a parallel statementfor right actions; we leave it to the reader to make the translation.

1.2 Some fundamental terminology

Let X, Y be (left) G-sets. A map φ : X−→Y is a G-map or a G-equivariant map if φ(gx) =gφ(x) for all g ∈ G, x ∈ X. Then G-sets and G-maps form a category that we will denote

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G-Set.

The fixed-point set XG is defined by XG = {x ∈ X : gx = x ∀g ∈ G}. This definesa functor G-set −→ Set in the evident way. The isotropy group Gx of a point x ∈ X isdefined by Gx = {g ∈ G : gx = x}; clearly Gx is a subgroup. Thus the fixed-points are thepoints x with Gx = G. The action is trivial if every point is a fixed point. At the oppositeextreme, the action is free if Gx = {e} for all x ∈ X.

Define an equivalence relation on X by x ∼ y if there exists g ∈ G such that gx = y. Aequivalence class is called an orbit, usually denoted O. The orbit determined by a particularx ∈ X is denoted Ox or Gx. The set of all orbits of a left action is denoted G\X; the setof orbits of a right action is denoted X/G. This notational distinction is important becausewe will often have groups acting on the left and the right of the same set X.

The action is transitive if there is only one orbit. In other words, for all x, y ∈ X thereexists a g such that gx = y. Transitive actions will be discussed in more detail in a latersection.

If O ⊂ X is an orbit and x ∈ O, then the map G−→X given by g 7→ gx factors through

a G-bijection G/Gx

∼=−→ O = Gx. Hence [G : Gx] = |Gx| (this is true even when the sets inquestion are infinite, but we have in mind here the finite case).

Finally, we recall a simple but very powerful counting formula. Suppose the group Gacts on the finite set X. Then

|X| =∑O|O| =

∑x∈G\X

[G : Gx],

where the first sum is over the orbits O of the action and the second sum means, in amildly abusive notation, that we are taking a fixed representative x of each orbit. Thischoice of x ∈ O is arbitrary, but the sum is nevertheless well-defined since [G : Gx] = |O| isindependent of the choice.

2 Examples

1. The symmetric group Sn acts on the left of [n] := {1, 2, ..., n} by permutions. The actionis transitive, with the isotropy group of any point isomorphic to Sn−1.

More generally, if X is any set, we let PermX denote the group of bijections X−→X.Then by construction PermX acts on the left of X by σ · x = σ(x). It is a left actionbecause a composition σ ◦ τ acts by τ first, then σ.

2. If G is any group, H any subgroup, then the left translation action of H on G isdefined by h · g = hg for h ∈ H, g ∈ G. The right translation action is given by g · h = gh.These are both free actions. The orbit space H\G of the left action is by definition the setof right cosets Hg, while the orbit space G/H of the right action consists of the left cosetsgH.

If G is finite, then (since every orbit has size |H|) the counting formula just says that|G| = |H| · [G : H].

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3. If G is any group, the left conjugation action of G on itself is given by g · x = gxg−1.Similarly right conjugation is defined by x · g = g−1xg. The fixed-point set of either actionis the center C(G) (Z(G) is another common notation for the center). The orbits are theconjugacy classes of G. The isotropy group of x is CGx, the centralizer of x in G.

In this case the counting formula yields the class equation. To state it we need a notationfor conjugacy classes, and—sadly—we have already assigned the letter “C” to centers andcentralizers. I will use the non-standard notation κ(x) to mean the conjugacy class of x, andConj G to mean the set of conjugacy classes. We then have

|G| =∑

x∈Conj G|κ(x)| =

∑x∈Conj G

[G : CGx].

Once again the notation has the obvious interpretation: We are choosing one x from eachconjugacy class, and the choice doesn’t matter.

4. If G is any group, let S(G) denote the set of subgroups of G. Then G acts on S(G)by left conjugation: g · H = gHg−1 (there is also a right conjugation, of course). Thefixed-points are the normal subgroups. The orbits are conjugacy classes of subgroups. Theisotropy group of H is the normalizer NGH of H in G.

5. Let X, Y be sets and let F (X, Y ) denote the set of functions X−→Y . If Y is a leftG-set, we get a left G-action on F (X, Y ) by (g · φ)(x) = g(φ(x)). If X is a left G-set, we geta right G-action on F (X, Y ) by (φ · g)(x) = φ(gx). Note carefully that this is a right action.However, we can always convert it to a left action by (g ? φ)(x) = φ(g−1x). If both X andY are G-sets, we can get a combined left action of G on F (X, Y ) by (g · φ)(x) = gφ(g−1x).

The fixed-point set of the “combined” left action gφg−1 is the subset of G-equivariantmaps X−→Y , as is easily checked.

6. Let X be a set, Xn the n-fold Cartesian product of X with itself. Then Sn acts onXn by permuting the coordinates. This is a right action, given explicitly by

(x1, ..., xn) · σ = (xσ(1), ...xσ(n)).

One could check directly that this is a right action, but easier is to note that Xn = F ([n], X),where Sn acts on the domain, on the left. So this is a special case of the previous example.The fixed-point set is the diagonal subset of all (x, x, ..., x). Contemplation of the orbits andisotropy groups is left to the reader.

7. Projective spaces. The purpose of this example is twofold. First of all, projectivespaces are ubiquitous in topology and geometry—including especially algebraic geometry,which leads me to discuss them in an algebra course. Second, in “nature” we are often firstconfronted not with a group action or even a group, but with a set (or topological space,etc.) X that may secretly be equipped with a useful group action and/or realization as theorbit set of a group action. It’s important to be able to recognize such structures.

Let F be a field, and let V be a finite dimensional vector space over F . (In fact thefinite-dimensionality isn’t necessary, but I prefer to avoid distractions.) The projective spaceP(V ) is the set of lines through the origin in V . Even though at the moment we are not giving

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it any topology, we call it a “space” because that is the traditional term, and if you call itthe “projective set” you risk being sneered at as an ignorant yokel from the backcountry.

Thus P(V ) does not involve a group in its definition, but it is in fact crawling with groups.First of all, it is the orbit set of the action of F× on V − {0} by scalar multiplication. Thisis a very handy interpretation. Second, GL(V ) acts transitively on it: By elementary linearalgebra, any line can be moved to any other line by an invertible linear transformation.This is a particularly important interpretation; it is often useful to recognize a set as atransitive G-set (also known as a “homogeneous space”). To complete the picture we choosea convenient point in the set and determine its isotropy group. Here there is no naturalchoice of a line; we just pick one and call it L0. The isotropy group H consists of invertibletransformations preserving L0, i.e. the set of A ∈ GL(V ) such that A has an eigenvector in

L0. Then there is a G-isomorphism GL(V )/H∼=−→ P(V ). If we want to be more explicit, we

choose a basis e1, ..., en and take L0 = 〈e1〉. Then H corresponds to the group of matriceswith ai1 = 0 for i > 1 (i.e. the first column is zero except for a11).

3 Actions preserving some additional structure

Frequently, the G-sets X one encounters are not merely sets but have some extra structurepreserved by the action. Some examples:

Example. X is itself a group, and G is acting on it via group automorphisms. In other words,g · (xy) = (g · x)(g · y) for all g ∈ G, x, y ∈ X. The action of G on itself by conjugation is anaction of this type. Actions via group automorphisms will be used to construct “semi-directproducts” later.

Example. We have a vector space V over a field F , and the G-action is linear: g · (v +w) =g · v + g · w, and g · (cv) = cg · v (c ∈ F ). This type of action is called a representation ofG over F. In this course “representation” will always be taken to mean finite dimensionalrepresentation, unless otherwise specified. Note that GL(V ) acts linearly on V by definition;we call this the standard representation of GL(V ).

Representation theory is one of the major branches of mathematics. We’ll considerrepresentation theory of finite groups in some detail, especially over C.

Example. Let F be a field, and suppose G acts on F via field automorphisms. This isprecisely the situation one studies in Galois theory.

Example. Let X be a topological space, and suppose G acts on X via homeomorphisms. Inother words, for each fixed g, the map x 7→ g · x is a homeomorphism. In fact we only needto check this map is continuous, since it automatically has an inverse given by the actionof g−1. In the topological case, however, G itself might be a topological group—i.e. both aspace and a group, with the multiplication and inverse maps continuous. In that context a“topological group action” means a group action such that the action map G ×X−→X iscontinuous (G×X has the product topology). This is usually a much stronger assumptionthan merely saying that G is acting by homeomorphisms.

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There are many variants on this theme: groups acting on metric spaces by isometries,smooth Lie groups actions on smooth manifolds (studied in our “Manifolds” course), sim-plicial actions on simplicial complexes...all of these are dear to my heart, but lie outside thescope an algebra course.

4 An alternate view of group actions

Let X be a G-set. Then each g ∈ G defines a bijection X−→X, so we get a map ρ :G−→PermX. Moreover, the axioms for a group action translate into the statement thatρ is a group homomorphism. Conversely if a homomorphism ρ : G−→PermX is given,we get a G-action on X by g · x = ρ(g)(x). Thus G-actions on X are the same thing ashomomorphisms G−→PermX. If ρ is injective we say that the action is faithul (“effective”is another commonly used term). Note that free actions are faithful, but not conversely: theaction of Sn on [n] is faithful, but certainly not free. Indeed an action is free if and onlyif all isotropy groups are trivial, whereas an action is faithful if and only if the intersection∩x∈XGx of all the isotropy groups is trivial.

In analyzing the structure of a newly encountered groupG, optimists hope to find a propernon-trivial normal subgroup H such that H and G/H are already “known”, or at least moreapproachable. Since kernels are always normal, Ker ρ is at least a candidate. Moreover, evenwhen Ker ρ is trivial (i.e. the action is faithul), thereby sinking the optimists’ strategy, weget an injective homomorphism G−→PermX that may yet prove useful for understandingG.

Let’s consider the case when G is finite and X is a finite G-set. If |X| = n, a choice ofordering of X yields an isomorphism PermX ∼= Sn, so we’ll think of ρ as a homomorphismG−→Sn. But to get any use out of this method we first need to find some finite G-sets.The most obvious candidate is G itself, with the left translation action. Since the actionis free and hence faithful, this yields an injective homomorphism ρ : G−→Sn. Thus everyfinite group is isomorphic to a subgroup of some Sn, a result known as “Cayley’s theorem”.The n obtained, however, is n = |G|, and |Sn| is too large for this result to be of more thanoccasional use.

To find smallerG-sets, we can choose a subgroupH and considerX = G/H with its trans-lation action. But how do we find subgroups of a general finite G? The Sylow p-subgroupsof G and their normalizers, discussed in the next section, form the most important and mostuseful family of subgroups that are defined for an arbitrary finite group. Alternatively, onemay be able to exploit special features of a particular G. As a simple, fun example, considerGL2F2. It acts linearly on F22, a vector space having a grand total of four elements. So thereare three nonzero vectors, and after choosing an ordering of them, we obtain a homomor-phism ρ : GL2F2−→S3. This ρ is especially useful, as it turns out to be an isomorphism(exercise).

Remarks. 1. If X is a G-set with extra structure, then ρ : G−→PermX lands in theautomorphism group of this structure. In example 1 we get ρ : G−→AutgrpX, in example 2we get ρ : G−→GL(V ), and so on. Indeed if C is any category, we can define a G-object inthe category to be an object X together with a homomorphism G−→AutCX.

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2. SupposeX1, X2 areG-sets with the same underlying setX but with differentG-actions.Let ρ1, ρ2 be the corresponding homomorphisms G−→PermX. Then X1 is isomorphic toX2 as a G-set if and only if ρ1, ρ2 are conjugate as homomorphisms to PermX. (Exercise.By “conjugate” I mean there is a σ ∈ PermX such that σρ1σ

−1 = ρ2. )

3. If X is a non-faithful G-set, with H = Ker ρ, then the G-action factors through aG/H-action: (gH)·x = gx. This is well-defined since H is normal. For example, consider theaction of GL(V ) on P(V ) discussed earlier. The kernel of this action is the scalar matricesF× ⊂ GL(V ), so we get an action of GL(V )/F× on P(V ). This is the reason GL(V )/F× iscalled the projective general linear group, denoted PGLnF .

5 Transitive G-sets

Recall that a G-set X is transitive if there is only one orbit, and that in this case a choice

of x ∈ X yields an isomorphism of G-sets G/Gx

∼=−→ X. Notice, however, that the choice ofx is arbitrary. This is true even for a free transitive action: then we get an isomorphism of

G-sets G∼=−→ X, where G has the left translation action, but there is no natural choice of

such an isomorphism; it depends on the choice of x.The next result is basic.

Proposition 5.1 Let X be a transitive G-set. Then the isotropy groups form a completeconjugacy class of subgroups of G.

Proof: Suppose x, y ∈ X. Choose g ∈ G with gx = y. Then gGxg−1 = Gy, as is readily

checked. Hence any two isotropy groups are conjugate. Conversely, let H be a subgroupconjugate to Gx; say gGxg

−1 = H. Then H = Ggx, so every subgroup in the conjugacy classoccurs as an isotropy group.

Now suppose we have two subgroups K,H and we want to show that K is conjugateto a subgroup of H, a problem that arises quite frequently. In the spirit of the previousproposition we have at once:

Proposition 5.2 K is conjugate to a subgroup of H if and only if the left action of K onG/H has a fixed point. More precisely, KxH = xH if and only if x−1Kx ⊂ H.

Now let’s consider the set of all orbits of the K-action on G/H. These are called the(K,H)-double cosets, denoted K\G/H. We could just as well think of K\G/H as the H-orbits of the right action onK\G, or more symmetrically as the orbits of the leftK×H-actionon G given by (k, h) · g = kgh−1. More often, however, we stick with the first interpretation.Here is an example known as the Bruhat decomposition:

Proposition 5.3 Let F be a field, and let B := BnF . Then the (B,B)-double cosets ofGLnF are given by

GLnF =∐w∈W

BwB,

where W ∼= Sn is the Weyl group.

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The proof is by old-fashioned row and column reduction, and is left to the reader. What’snot obvious is why this particular way of arranging the row/column reduction is of interest. Itturns out that it plays an important role in the structure theory of a large class of interestinggroups (not just the general linear groups), and enters into algebraic geometry and topologyvia “Schubert varieties” and “Schubert cells”. For example, when F = R or F = C then theorbit set GLnF/B is a topological space known as a “flag manifold” or “flag variety”, and theleft B-orbits—which by the proposition are indexed by elements of W—are homeomorphicto vector spaces over F and called Schubert cells. They are very useful for studying thegeometry and topology of flag manifolds, a subject which is in itself a major industry thesedays. I mention all this just to pique your curiosity, and to suggest how the humble processof row reduction connects with beautiful, deep mathematics.

6 A fixed-point theorem, and the Sylow theorems

Throughout this section, p is a prime. Everything will be based on the following simplefixed-point theorem:

Theorem 6.1 Let P be a finite p-group, S a finite P -set. Then |S| = |SP | mod p.

Proof: |S| =∑ |O|, where O ranges over the P -orbits. Since P is a p-group, |O| is either

divisible by p or consists of a single fixed point, whence the result.

Corollary 6.2 If |S| is prime to p, then there is at least one fixed point.

As an immediate application of the corollary, we have:

Proposition 6.3 Every finite p-group G has non-trivial center.

Proof: Consider the action of G by conjugation on G − {e}. By the corollary it has afixed-point, so G has non-trivial center.

The proposition in turn has a nice corollary:

Corollary 6.4 If G has order p2 (p a prime) then G is abelian.

Proof: By the proposition, G has a central subgroup H of order p. Hence |G/H| = p, soG/H is cyclic. But whenever a group G has a central subgroup with cyclic quotient group,G is abelian (why?).

The next application can be proved using the binomial theorem, but we can also deduceit from Theorem 6.1.

Proposition 6.5 Let n = spk. Then(npk

)= s mod p.

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Proof: Partition [n] into s disjoint subsets A1, ..., As of size pk. Let Ci ⊂ Sn be a cyclicgroup of order pk that permutes Ai transitively and fixes the other Aj’s pointwise, and takeP = C1 × ... × Cs. Then the action of P on Ppk [n] has exactly s fixed points, namelyA1, ..., As. Hence the proposition follows from the theorem.

Let G be a finite group, and write |G| = spk with s prime to p. A p-Sylow subgroup isa subgroup P of order pk. The first Sylow theorem asserts that such subgroups alwaysexist.

Theorem 6.6 G has a p-Sylow subgroup.

Proof: The strategy is to look for a finite G-set S such that some isotropy group Gx is ap-Sylow subgroup. Indeed, suppose we had a finite G-set S such that (i) |S| is prime to p;and (ii) all isotropy groups Gx are p-groups. Then by (i) there is an orbit O with |O| primeto p. Choose x ∈ O; then |Gx| = pi for some i by (ii). But |G| = |Gx| · |O|, forcing i = k.Hence Gx is a p-Sylow subgroup.

It remains to exhibit such an S. Take S to be the set of subsets of G of size pk, withaction induced by the left translation action of G on itself. Then (i) holds by Proposition 6.5.Now let A ∈ S. Then GA acts freely on the elements of A (since the left translation actionis free), so |GA| divides pk, proving (ii).

The second Sylow theorem says that all p-Sylow subgroups are conjugate. Moreprecisely:

Theorem 6.7 Let H be a p-subgroup of G, P a p-Sylow subgroup. Then H is conjugate toa subgroup of P . In particular, any two p-Sylow subgroups are conjugate.

Proof: As discussed earlier, this is equivalent to saying that the action of H on G/P hasa fixed point: if HxP = xP , then x−1Hx ⊂ P and conversely. But H is a p-group and|G/P | = s is prime to p, so this follows immediately from Theorem 6.1.

The last item of business is to say something about the set of all p-Sylow subgroups ofG. How many such subgroups are there? By the second Sylow theorem, G acts transitivelyon this set by conjugation. If we fix a p-Sylow subgroup P , the isotropy group of the actionis the normalizer NGP . Hence the number of p-Sylow subgroups is [G : NGP ]. This bringsus to the third Sylow theorem:

Theorem 6.8 Let npG denote the number of distinct p-Sylow subgroups of G. Then npGdivides |G| and npG = 1 mod p.

Proof: Since npG = [G : NGP ] for any choice of p-Sylow subgroup P , npG divides |G|.Now fix a p-Sylow subgroup P , and note that by the second Sylow theorem P is the uniquep-Sylow subgroup of NGP (since it is a normal p-Sylow subgroup of NGP ). Now consider theleft translation action of P on G/NGP . If PxNGP = xNGP then x−1Px ⊂ NGP , forcingx−1Px = P since P is the unique p-Sylow subgroup of NGP . Hence x ∈ NGP ; in otherwords, the P -action has a unique fixed point. Using Theorem 6.1 we conclude

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npG = |G/NGP | = (G/NGP )P = 1mod p.

Here’s another interesting fact about p-Sylow subgroups:

Proposition 6.9 Let P be a p-Sylow subgroup of G, and suppose NGP ⊂ H. Then NGH =H. (In particular, this is true for H = NGP .)

Proof: Suppose gHg−1 = H. Then gPg−1 is a p-Sylow subgroup of H, so by the secondSylow theorem there is an h ∈ H such that hPh−1 = gPg−1. Then h−1g ∈ NGP , so g ∈ H.

Many examples and applications of p-Sylow subgroups can be found in the exercises.

7 New G-sets from old: restriction, disjoint unions,

products, and induction

Change of group and restriction. Suppose X is a left G-set and φ : H−→G a homomorphism.Then X is a left H-set with action h · x = φ(h)x. Thus φ defines a functor φ∗: G-set −→H-set. The case when φ is inclusion of a subgroup is of particular importance; in this case weuse the notation XH for X regarded as an H-set, and call X 7→ XH the restriction functor.

Disjoint unions. Suppose X and Y are G-sets. The disjoint union X∐Y is a G-set in the

obvious way, and in fact is the categorical coproduct. Similarly if Xα is any collection of G-sets indexed by a set J ,

∐α∈J Xα is a G-set and is the categorical coproduct. The fixed-point

set and orbit set functors take coproducts of G-sets to coproducts of sets.

Products of G-sets. Again let X, Y be G-sets. The product X×Y is a G-set via the diagonalaction: g · (x, y) = (gx, gy). The product of any collection of G-sets is defined similarly,and is the categorical product. The fixed-point functor takes products to products, e.g.(X × Y )G = XG × Y G. Orbits, however, are another matter; there is no simple relationshipbetween G\(X × Y ) and G\X, G\Y . We will see some examples later.

Balanced products. Suppose X is a right G-set, Y a left G-set. Define an equivalence relationon X×Y by (xg, y) ∼ (x, gy). The balanced product X×GY is the set of equivalence classes.In fact this is just the orbit set of the left G-set X×Y , where g acts by g ·(x, y) = (xg−1, gy).But the slight change in viewpoint can be useful and enlightening. For our immediatepurposes, the “induced” G-spaces below provide the most important example.

Induction. Suppose H is a subgroup of G and X is a left H-set. Then the balanced productG×HX is a left G-set with action g1 ·[g, x] = [g1g, x]; we say that the G-action is induced fromthe H-action. Here the brackets [] denote equivalence class in the balanced product. ThusX 7→ G×H X defines the induction functor H-set −→ G-set (the definition of the functoron morphisms being obvious). Note that the map i : X−→G×H X given by i(x) = [e, x] isan H-map.

Induction has the following universal property (see the category theory notes for a generaldiscussion of such properties). We keep the above notation.

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Proposition 7.1 Suppose Y is a G-space, X is an H-space and φ : X−→Y is an H-map.Then there is a unique G-map ψ : G×H X−→Y such that the following diagram commutes:

X G×H X

Y

-i

?

φ

ppppppppppp ∃!ψ

Proof: There is no choice in the definition of ψ: We must take ψ([g, x]) = gφ(x). Now checkthat it works (in particular, check that ψ is well-defined).

As usual, we give two alternate ways of thinking about the universal property:

Plain English version. If you want to define a G-map G ×H X−→Y , it is enough (indeedequivalent) to define an H-map X−→Y .

Adjoint functor version. Induction H-set −→ G-set is left adjoint to restriction G-set −→H-set.

See the category theory notes for discussion of adjoint functors. In essence there is notmuch to it in the present example; the universal property translates immediately to theassertion that there is a bijection

HomG(G×H X, Y ) ∼= HomH(X, Y ).

To complete the proof that these are adjoint functors, one has to show that the abovebijection is natural in X and Y . This is easy once one has absorbed the definition of“natural transformation”, but it is not essential to understand all this right away. Just usethe universal property.

There is a recognition principle for induced G-sets. Let Y be a G-set, X ⊂ Y an H-invariant subset (H ⊂ G). Applying the universal property to the inclusion j : X−→Y , weget a G-map ψ : G×H X−→Y ; indeed it is just ψ([g, x]) = gx.

Proposition 7.2 ψ is bijective if and only if (i) for all y ∈ Y , there is a g ∈ G such thatgy ∈ X; and (ii) whenever x1, x2 ∈ X and gx1 = x2, we have g ∈ H.

The proof is a straightforward check; (i) gives the surjectivity and (ii) the injectivity.Note that (ii) says that if g /∈ H, then g moves every element of X to an element not in X.

Example. Let F be a field, G = GLnF , and Y the set of pairs (L, v) with L a line in F n andv ∈ L. We have an evident G-action on Y given by g · (L, v) = (gL, gv). Let L0 denote theline spanned by the standard basis vector e1, and let X = {(L, v) ∈ Y : L = L0}. Then Xis invariant under H := {g ∈ GLnF : gL0 = L0}. Now let’s check conditions (i) and (ii) ofthe recognition principle above: (i) is clear, since by linear algebra GLnF acts transitively

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on the lines. And if (gL0, gv) = (L0, w) then g ∈ H. So we have a canonical isomorphism

of G-sets G ×H X∼=−→ Y . To complete this description, one should describe how H acts

on X. Identifying X with the vector space L0, it is the linear action that pulls back scalarmultiplication along the homomorphism H−→F× taking a matrix A to a11, the upper leftentry: A · (L0, v) = (L0, a11v).

Examples of this type occur frequently in geometry and topology.

8 Semi-direct products and group extensions

8.1 Prelude on products

One of the most commonly used techniques in mathematics—and here I admit to stating theobvious, but bear with me—is to combine simple objects in some way to build complicatedobjects, and conversely to understand complicated objects by breaking them down intosimpler objects. In the case of groups, for example, given groups H,K we can form a newgroup by taking the product H ×K. Conversely, if we can decompose a given group G as aproduct G ∼= H×K, where H and K are already understood, then—at least in principle—wecan understand G. To carry out this latter strategy we need a recognition principle for suchproducts. The reader has probably already seen this, but here is a reminder of how it worksfor an arbitrary finite number of factors:

Proposition 8.1 Suppose H1, ..., Hn are normal subgroups of G such that (i) Hi∩(∏j 6=iHj) =

{e} and (ii) G = H1...Hn. Then the multiplication map m : H1 ×H2 × ...×Hn−→G is anisomorphism of groups.

Proof: First note that m is a group homomorphism: For this, one needs to know that fori 6= j the elements of Hi, Hj commute with one another. But if x ∈ Hi and y ∈ Hj, thenby normality xyx−1y−1 ∈ Hi ∩Hj = {e}. Then m is injective by (i) and surjective by (ii),completing the proof.

8.2 Semi-direct products

But only if we are lucky will G decompose as a product. The next best thing is a “semi-direct product”, which we know describe. Suppose given groups H,K and a homomorphismρ : K−→AutH. Equivalently, we are given a left action of K on H by group automorphisms.Associated to this data we have the semi-direct product H oρ K: As a set it is just H ×K,but with group multiplication defined by

(h1, k1) · (h2, k2) = (h1(ρ(k1)(h2)), k1k2).

The proof that this multiplication defines a group structure is left as an exercise. Notethat (h1, e) · (e, k2) = (h1, k2). So there is no harm in dropping the parentheses and writinghk in place of (h, k). Note also that the outer two factors in the displayed product just goalong for the ride; all the action takes place with the inner two. Thus the essence of themultiplication rule is that it tells you how to commute k with h: kh = (ρ(k)(h))k. But we

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already know how to do this, in any group: kh = (khk−1)k. The upshot of this discussionis that in the semi-direct product, the automorphism ρ(k) of H is the same thing as leftconjugation by k.

Note that by construction H is normal in H oρK, and that H oρK is the direct productH ×K if and only if ρ is trivial if and only if K is normal (check this!).

If ρ is understood, we often omit it from the notation and simply write H oK. Needlessto say, abusive notation of this kind must be used with care.

Example. Suppose we ask: Are there non-abelian groups of order 21? With semi-directproducts in hand, it is easy to construct such groups explicitly: AutC7 is cyclic of order6, so we can choose an injective homomorphism ρ : C3−→AutC7 (there are two such ho-momorphisms) and form C7 oρ C3. More generally, given primes p, q with p|q − 1, we getnon-abelian groups of order pq this way. Indeed one can show that every group of order pqhas this form.

More importantly, many groups occuring “in nature” can be recognized as semi-directproducts and thereby better understood. Here is a simple recognition principle:

Proposition 8.2 Suppose G contains subgroups H,K such that:(a) H is normal in G.(b) H ∩K = e.(c) HK = G.Then G = H oρ K, where ρ(k)(h) = khk−1.

Proof: Conditions (b),(c) imply that multiplication H ×K−→G is a bijection. Since H isnormal, the stated homomorphism ρ is defined, and a trivial check (which we have in effectalready done above) shows that the multiplication on G is exactly the semi-direct productmultiplication.

With this criterion in hand, you soon realize that semi-direct products are everywhere.Here are a few important examples, with details and verifications left to the reader:

Examples. 1. The dihedral group of order 2n is a semi-direct product Cn o C2, where C2

acts on Cn as multiplication by −1.2. Sn = An o C2.3. Let V be a finite dimensional vector space over a field F . The affine group Aff(V ) is

defined to by the subgroup of PermV generated by GL(V ) and the subgroup of translationsTv : x 7→ x + v . The group of translations is isomorphic to the additive group of V , andAff(V ) = V o GL(V ). Here the action of GL(V ) on V by conjugation is the same as thestandard action. When V = F n, we also write AffnF in place of Aff(F n).

4. Consider the symmetric group Sp and the p-Sylow subgroup Cp generated by thestandard p-cycle (12...p). Then the normalizer NSpCp is isomorphic to Aff(Fp) and hence isa semi-direct product of the form Cp o Cp−1.

5. BnF = UnF oDnF . What is the action of DnF on UnF?6. NnF = DnF o Sn, where Sn acts on DnF = (F×)n by permuting the factors. This is

a type of semi-direct product known as a wreath product, disussed further in the exercises.

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8.3 Group extensions

A group extension consists of group homomorphisms

Hi−→ G

π−→ K,

where π is surjective and i is an isomorphism onto the kernel of π. Thus without loss ofgenerality we can, if desired, assume that i is just an inclusion and K = G/H. In fact wewill often treat H as a subgroup and omit i from the notation.

Note that H and K alone do not determine G, even when G is abelian. For example, ifwe are given a group extension C2−→G−→C2, then since |G| = 4 we know G is abelian, butwithout further information there is no way to know whether G is C4 or C2 × C2.

The extension is central if H ⊂ C(G). Note, for example, that if H = C2 then theextension is automatically central. This leads to another example of the ambiguity inherentin group extensions: If we have an extension C2−→G−→C2×C2, then it is a central extensionbut G could be any of the five groups of order 8 except C8: (C2)

3, C2 × C4, the dihedralgroup D8, or the quaternion group Q8. Each of these four groups fits into such an extension,as you can easily check.

Note that a semi-direct product G := H o K fits into an extension H−→G−→K. Infact we can characterize the semi-direct products in terms of extensions. A group extension

Hi−→ G

π−→ K splits if there is a homomorphism s : K−→G such that π ◦ s = IdK . Wecall s a splitting (or sometimes a “section”) of π.

Proposition 8.3 If G := HoK is a semi-direct product, the extension H−→G−→K splits.Conversely if H−→G−→K is a split group extension, then G ∼= HoK with K acting on H byconjugation. More precisely, if s : K−→G is a splitting, K acts on H by k ·h = s(k)hs(k)−1.

Proof: If G = H oK, define s by s(k) = (e, k). Conversely if H−→G−→K is split, choose asplitting s : K−→G. Then the pair H, s(K) satisfies the recognition principle for semi-directproducts (an easy check).

Remark. Note that to give a splitting s : K−→G is the same thing as giving a subgroupK ′ ⊂ G such that π : G−→K maps K ′ isomorphically to K.

Example. Let G be a group of order pq, where p, q are primes with p < q. I claim that G isa semi-direct product of the form Cq o Cp. First of all, by the third Sylow theorem there isa unique and hence normal q-Sylow subgroup, cyclic of order q. Hence there is an extensionCq−→G

π−→ Cp. Now choose a p-Sylow subgroup H. Then π|H is injective, and hencean isomorphism. This proves the claim. Note that Cp acts on Cq by some homomorphismCp−→AutCq ∼= Cq−1, and hence for the action to be non-trivial we must have p|(q − 1) orequivalently q = 1 mod p. This fits with the third Sylow theorem because if H is not normal,then there are q p-Sylow subgroups.

9 Solvable and nilpotent groups

In attempting to analyze a group G by fitting it into an extension H−→G−→K—or equiva-lently, finding a normal subgroup H with quotient K—one simple possibility we might hope

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for is to find such an extension with both H and K abelian. Or if we are feeling especiallylucky, we might hope that in addition the extension is central. A more reasonable althoughstill optimistic hope is that we can build G by a finite iteration of such extensions. Thisleads to the concepts solvable and nilpotent group in the respective cases.

Example. We can build S4 in two steps out of abelian groups. First we form the extensionC2

2−→A4−→C3. Then we form the extension A4−→S4−→C2.

But we need a smoother way to think about “building up from extensions”. This is thesubject of the next section.

9.1 Don’t fight it, filter it!

An increasing filtration of a group G consists of subgroups

{e} ⊂ G1 ⊂ G2 ⊂ G3....

A decreasing filtration likewise consists of subgroups

G ⊃ G1 ⊃ G2 ⊃ G3 ⊃ ...

In either case the filtration stabilizes at Gn if Gk = Gn for all k ≥ n. An increasing (resp.decreasing) filtration is finite if Gn = G for some n (resp. Gn = {e} for some n). For finitefiltrations there is no real difference between the increasing and decreasing cases, since onecould always reverse the ordering to convert from one to the other.

In fact this definition makes sense for any kind of object with subobjects: rings andsubrings, vector spaces and sub-vector spaces, topological spaces and subspaces, etc. Ingroup theory filtrations are classically known as “series”. I prefer the term “filtration”because it has a verb, “to filter”, that goes with it, and because it is the more widely usedterm across many different categories. But I will freely use both terms, just so you get usedto them.

Example. Let p be a prime. Any abelian group A has a natural decreasing “p-adic” filtrationA ⊃ pA ⊃ p2A ⊃ ... (which need not terminate; think of A = Z), as well as a natural p-torsionfiltration A[p] ⊂ A[p2] ⊂ ... (which need not terminate; think of A = Q/Z). Incidentally, itisn’t necessary for p to be a prime here, but the prime case is by far the most important.

Example. Note that according to our definition, a filtration of a finite group need not bea finite filtration. For example, suppose p, q are distinct primes, and G is a finite abelianq-group. Then pG = G and hence the p-adic filtration takes the form G ⊂ G ⊂ G ⊂ ...; itnever reaches {e}. Similarly the p-torsion filtration takes the form {e} ⊂ {e} ⊂ ...; it neverreaches G. Thus a finite filtration is not merely one with a finite number of distinct terms,but one that begins at the trivial subgroup and ends at the whole group (or vice-versa).

Usually the point of filtering a group (or anything else) is to arrange it in such a waythat the quotient objects Gk/Gk−1 (increasing case) or Gk/Gk+1 (decreasing case) havesome simple form that we understand; then the hope is that we can recover, perhaps by an

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induction argument, information about G itself. This is the meaning of the motto: Don’tfight it, filter it! In our category of groups, as it stands the quotients are in general onlysets, so we make the definitions:

Definition. An increasing filtration is subnormal if Gi is normal in Gi+1 for all i, and normalif Gi is normal in G for all i. Thus normal implies subnormal but not conversely (for aminimal counterexample to the converse, look at our old friend A4). Subnormal and normaldecreasing filtrations are defined similarly. The classical terminology is “subnormal/normalseries”.

It will be handy to have a term for the following simple construction: SupposeH−→G π−→K is a group extension, and we are given finite filtrations (which we can assume are increas-ing) of H and K. Then we get a filtration of G by “splicing” them together:

{e} = H0 ⊂ H1 ⊂ ... ⊂ H ⊂ π−1K1 ⊂ ...π−1Kn = G.

We call this the splice of the two filtrations. Note that the splice of subnormal filtrations issubnormal, and that the list of quotient groups obtained is just the union of the quotientgroups of the original filtrations.

Caution: The splice of normal filtrations need not be normal. Once again, A4 provides acounterexample. In order for the splice to be a normal filtration, the filtration on H wouldhave to be invariant under the conjugation action of K on the set of normal subgroups of H.

Now, let’s get on to our main examples.

9.2 Solvable groups

Abelian groups are easier to understand than general groups. In the spirit of our filtrationmotto, therefore, it is reasonable to make the following definition: A group G is solvable ifit admits a finite subnormal filtration with abelian quotients. (The terminology comes fromGalois theory, where group theory originated.) We call such a filtration a solvable filtration(or series) for G. Any abelian group is solvable, and if H−→G−→K is a group extensionwith H,K abelian, then G is solvable. Before giving further examples, we establish somebasic properties of solvable groups.

First of all, there is a “functorial” decreasing filtration with abelian quotients of any groupG, defined recursively by G0 = G and Gi+1 = [Gi, Gi]. We call this the commutator filtration.Note that it is not only a normal filtration, but even a characteristic filtration; i.e. thesubgroups in question are characteristic subgroups (invariant under arbitrary automorphismsof G). The first thing to note is that it has limited applicability; indeed for many groups itis a useless filtration: For example, if G is a nonabelian simple group then it is the filtrationG ⊃ G ⊃ G.... But if it does yield a finite filtration, i.e. one that reaches {e}, then G issolvable. We will prove the converse shortly.

Remark. By “functorial” we mean that any group homomorphism preserves the filtration. Tomake this fit precisely into the framework of category theory, we would define a category offiltered groups, with morphisms the filtration preserving homomorphisms, so that assigning

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to G its commutator filtration is a functor to this new category. But there’s no compellingreason to do this at the moment.

Proposition 9.1 The following are equivalent:a) The commutator filtration of G is finite, i.e. G(m) = {e} for some m.b) G admits a finite normal filtration with abelian quotients.c) G is solvable.

Proof: Clearly (a) ⇒ (b) ⇒ (c). Now suppose G is solvable, and let G = H0 ⊃ H1 ⊃ ... ⊃Hm = {e} be a solvable filtration. Then G/H1 is abelian, so [G,G] ⊂ H1. Similarly, sinceH1/H2 is abelian, we have G(2) ⊂ [H1, H1] ⊂ H2. Continuing in this manner, we find thatG(k) ⊂ Hk for all k. Hence G(m) = {e}, proving that (c) ⇒ (a).

Proposition 9.2 Solvable groups are closed under taking subgroups, quotients, extensions,and finite products.

Proof: We first show that if G is solvable, so is any quotient group. Suppose π : G−→His a surjective homomorphism. Let G ⊃ G1 ⊃ G2 ⊃ ... be a solvable filtration for G, andconsider the filtration π(Gi) of H. It is a finite subnormal filtration, and the quotients areabelian because π(Gi)/π(Gi+1) is a quotient of Gi/Gi+1. Hence H is solvable. The case ofsubgroups is equally straightforward, and left to the reader.

Next we show that solvable groups are closed under extensions; that is, if H−→G−→Kis a group extension and H,K are solvable, so is G. This is immediate because givensolvable filtrations for H,K we can splice them to get a solvable filtration for G (recall thatsubnormality is preserved under splicing, and the quotients remain abelian because they arein fact identical to the original quotients).

Finally, consider products. The product of two solvable groups G,H is solvable becauseof the extension H−→G×H−→G. Induction on the number of factors then shows that anyfinite product of solvable groups is solvable.

Remark: It follows that the solvable groups can be described as the smallest class of groupsthat contains the abelian groups and is closed under extensions.

Now, here is one of the most important solvable groups. We could work over a generalcommutative ring, but to avoid distractions we will stick to the case of a field F . Recall thatBnF ⊂ GLnF is the Borel subgroup of upper triangular matrices.

Proposition 9.3 BnF is solvable.

Proof: Note thatB1F = F×, and that there is a surjective homomorphism π : BnF−→Bn−1Fgiven by simply projecting A ∈ BnF onto its upper left (n−1)× (n−1) block. By inductionwe can assume Bn−1F is solvable, so it suffices to show Ker π is solvable. Now Ker π isthe “right column group” consisting of matrices that equal the identity in the first n − 1columns, which we will quaintly denote RCnF . It fits into an extension RCu

n−→RCn−→F×,where the second map is just projection on the nn coordinate and RCu

n is therefore the

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subgroup with ann = 1 (the superscript u if for “unipotent”). One easily checks that RCun

is isomorphic to the additive group F n−1, and in particular is abelian. So RCnF is solvableand the proof is complete.

See the exercises for a discussion of the commutator series of BnF .

The finite p-groups are another important family of solvable groups. However, theysatisfy the even stronger property of “nilpotence”, as we will show in the next section.

There are a number of theorems showing that under certain restrictions on the primefactors of n, every group of order n is solvable. Here are three such theorems, in increasingorder of difficulty (the first is by far the easiest, and is demoted to the status of “proposition”):

Proposition 9.4 a) If n = pq with p, q prime, then every group of order n is solvable.b) If n = pqr with p, q, r distinct primes, then every group of order n is solvable.

Proof: Exercise. Part (a) is trivial from things we’ve already proved. Part (b) is a littlemore interesting.

The next result is known as Burnside’s paqb theorem.

Theorem 9.5 Suppose p, q are primes and |G| = paqb. Then G is solvable.

The proof is a beautiful application of representation theory, as we will show later.

Corollary 9.6 Every group of order < 60 is solvable.

Proof: 60 = 22 · 3 · 5 is the smallest number that is neither the product of three distinctprimes nor of the form paqb. (It’s also a nice exercise to prove the corollary directly, withoutBurnside’s theorem.)

The next theorem is due to Feit and Thompson in 1963.

Theorem 9.7 Every finite group of odd order is solvable.

The original proof is 200 pages long, and as far as I know has never been simplified.

9.3 Nilpotent groups

An increasing normal filtration {e} = G0 ⊂ G1 ⊂ ... of G is central if Gi+1/Gi ⊂ C(G/Gi).We say that G is nilpotent if admits a finite central filtration, i.e. one that terminates at G.Note that nilpotent groups are solvable, since any central filtration is a solvable filtration.On the other hand, S3 is solvable but not nilpotent.

Proposition 9.8 Every finite p-group is nilpotent.

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Proof: Let G be a finite p-group. Then C(G) is non-trivial. By induction on order we canassume G/C(G) has a finite central filtration; splicing with the one-step filtration {e} ⊂C(G) yields the result.

In fact any group has a functorial, normal central filtration, sometimes called the as-cending central series, defined recursively as follows: Let C1 = C(G). Having definedthe normal subgroup Ck, let π : G−→G/Ck be the quotient homomorphism, and setCk+1 = π−1C(G/Ck). It is clear that Ck+1 is normal.

The next proposition is analogous to Proposition 9.1.

Proposition 9.9 The following are equivalent:a) The ascending central series of G is finite; i.e., ends at G.b) G is nilpotent.

Proof: Clearly (a) ⇒ (b). To show that (b) ⇒ (a), assume given a central filtration Gi withGm = G and show inductively that Gi ⊂ Ci. Then Cm = G, as desired.

Before proceeding further, it will be convenient to generalize the concept “nilpotentgroup” to nilpotent group action. Suppose K,G are groups and K acts on G by groupautomorphisms. We say that the action in nilpotent if G has a finite subnormal filtration{e} ⊂ G1... ⊂ Gm = G such that (i) each Gi is invariant under the K-action, and (ii) theinduced K-action on each quotient Gi/Gi−1 is trivial. We call such a filtration K-nilpotent.

Example. Let G act on itself by conjugation. This action is nilpotent if and only if G isnilpotent.

Example. Consider a field F and take K to be the group of upper triangular unipotentmatrices UnF . For G we take the additive group F n, with its standard left UnF action.Filter F n by the F i’s (where as always, our default inclusion F i ⊂ F n is in the first icoordinates). This is automatically a subnormal (indeed normal) filtration, since F n isabelian, and satisfies the nilpotent action conditions (i)-(ii) by definition of UnF .

Note: From the point of view of this example, it would have made more sense to use theterm “unipotent action” in place of “nilpotent action”. But such terminology conflicts areinevitable, and one just has to live with them.

Nilpotent groups are not closed under extensions (think of S3, for example). Our nextdefinition compensates for this deficiency: A group extension H−→G−→K is nilpotent if theconjugation action of G on H is nilpotent. Any central extension is nilpotent, for example,or more generally any extension in which the conjugation action of G on H is trivial. Theextension C3−→S3−→C2 is not nilpotent.

Proposition 9.10 The class of nilpotent groups is closed under subgroups, quotients, nilpo-tent extensions and finite products.

Proof: Let G be nilpotent, H a subgroup. Since the action of G on itself by conjugation isnilpotent, so is the restriction of this action to H, i.e. the action of H on G by conjugation.

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But H is invariant under the latter action, and we get a finite central filtration of H byintersecting with a given such filtration for G. The case of quotients is also straightforward,and left to the reader.

Now suppose H−→G−→K is a group extension with K nilpotent and G acting nilpo-tently on H by conjugation (which implies that H is nilpotent). Choose a G-nilpotentfiltration of H and a nilpotent filtration of K; splicing these yields a nilpotent filtration ofG, as desired. Details are left to the reader.

If H,G are nilpotent, then H−→G × H−→G is clearly a nilpotent extension, since theconjugation action of G × H on H factors through H. So H × G is nilpotent. It thenfollows by induction on the number of factors that any finite product of nilpotent groups isnilpotent.

Next is one of the most important examples. Let F be a field, and recall that theunipotent group UnF is the group of upper triangular n×n matrices with 1’s on the diagonal.Since it is a subgroup of the solvable group BnF , it is solvable. But more is true:

Proposition 9.11 UnF is nilpotent.

Proof: We follow closely the proof already given for solvability of BnF . As in that case,there is a group extension

RCunF−→UnF−→Un−1F,

where the second map is projection on the upper left (n− 1)× (n− 1) block and RCunF is

again the “unipotent right column group”; for example RC3F consists of matrices 1 0 a0 1 b0 0 1

By induction we can assume Un−1F is solvable, so it suffices to show that the action of UnFon RCu

nF by conjugation is nilpotent. Since RCunF is abelian, the action of UnF factors

through Un−1F , so what we need to show is that the conjugation action of Un−1F on RCunF

is nilpotent. But under the evident isomorphism RCunF∼= F n−1 (where the entries of the

right column are ordered from top to bottom), this action corresponds to the standard linearaction of Un−1F on F n−1. We saw earlier that this latter action is nilpotent, so the proof iscomplete.

Here are two more interesting facts about nilpotent groups. Both are false for solvablegroups; the reader can easily supply examples.

Proposition 9.12 Let G be a nilpotent group, H a normal subgroup. Then H∩C(G) 6= {e}.

Proof: Since G acts nilpotently on itself by conjugation, and H is invariant, it acts nilpotentlyon H. In particular H has a non-trivial subgroup H1 on which G acts trivially, so H1 ⊂H ∩ C(G).

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Proposition 9.13 Let G be a nilpotent group, H a proper subgroup of G. Then H is aproper subgroup of its normalizer NGH.

Proof: If C := C(G) is not contained in H, then H 6= CH ⊂ NGH. If C ⊂ H, then byinduction we can assume H/C 6= NG/CH/C; it follows at once that H 6= NGH.

Now we can neatly characterize finite nilpotent groups.

Theorem 9.14 A finite group G is nilpotent if and only if G is a product of p-groups (wherep ranges over the prime divisors of |G|).

Proof: We have shown that any p-group is nilpotent. Since any finite product of nilpotentgroups is nilpotent, this proves the “if”.

Conversely, suppose G is nilpotent, let p divide |G|, and let P be a p-Sylow subgroup.Then NG(NGP ) = NGP (see the section on the Sylow theorems). By Proposition 9.13this is impossible unless NGP = G, so P is normal. So if p1, ..., pm are the prime divisorsof G, with corresponding unique pi-Sylow subgroups Pi, the natural multiplication mapφ : P1 × ...× Pm−→G is an injective group homomorphism. Comparing orders, we see thatit must be an isomorphism.

The descending central series of a group G is defined recursively by C0 = G and Ck+1 =[G,Ck]. Thus C1 = [G,G], so this starts off the same as the commutator series. But atthe next step we take [G, [G,G]] instead of [[G,G], [G,G]]. By construction the descendingcentral series is a characteristic (hence normal) filtration, whose quotients are not onlyabelian but satisfy Ck/Ck+1 central in G/Ck+1. Hence if the descending central series isfinite, i.e. ends at {e}, reversing the order yields a finite central increasing filtration of G.This proves one direction of the following proposition; the converse is left to the reader.

Proposition 9.15 G is nilpotent if and only if the descending central series is finite, i.e.Cn = {e} for some n.

We conclude by mentioning a class of finite groups lying between the nilpotent andsolvable groups. If G is a finite group, G is supersolvable if it has a finite normal filtrationwith cyclic quotients.

Proposition 9.16 For finite groups G,nilpotent ⇒ supersolvable ⇒ solvable.

The second implication is immediate, while the first is left as an exercise. Note thatS3 is supersolvable but not nilpotent, while A4 is solvable but not supersolvable. One canalso check that any solvable finite group admits a subnormal filtration with cyclic quotients;hence the insistence on normal filtrations in the definition of supersolvable groups is key.Supersolvable groups have nice representation-theoretic properties, as we will see in [Serre],§8.5.

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10 Simple groups and perfect groups

10.1 Simple groups

A group G is simple if it has no non-trivial proper normal subgroups. If G is abelian, then Gis simple if and only if it is cyclic of prime order (an easy exercise). Non-abelian finite simplegroups were not completely classified until 2004; the proofs occupy thousands of pages. Theformidable Feit-Thompson theorem implies that every non-abelian finite simple group haseven order, but this is just the first little step!

Note that since every group of order < 60 is solvable, there are no non-abelian simplegroups of order < 60 (indeed these two statements are equivalent).

Proposition 10.1 The alternating group An is simple for n ≥ 5.

For a short and interesting proof, see Artin’s undergraduate algebra text (proofs can befound also in [Hungerford] and [Dummit-Foote]). Another infinite family of simple groupsis given by the projective special linear groups PSLnF , F a field. These are defined byPSLnF = SLnF/C, where C := C(SLnF ) is the center. Since C(GLnF ) consists of thescalar matrices, it is easy to see that C(SLnF ) is also just the scalar matrices with deter-minant 1, and hence is a finite cyclic group of order dividing n. It turns out that with justtwo exceptions, PSLnF is simple for all n, F . We’ll prove this in a later section for n = 2,and sketch the proof for general n. By taking F to be a finite field we get an infinite familyof finite simple groups. This is as far as we’ll go.

10.2 The Jordan-Holder theorem

Let G be a group. A Jordan-Holder filtration of G is a finite increasing subnormal filtrationwith simple quotients. Such a filtration need not exist; for example, an abelian group admitsa Jordan-Holder filtration if and only if it is finite (easy exercise). On the other hand, clearlyany finite group admits a Jordan-Holder filtration.

Here is the Jordan-Holder theorem:

Theorem 10.2 If G admits a Jordan-Holder filtration, then the list of simple groups occur-ing as the quotients is independent of the choice of filtration, up to ordering.

We will rarely—if ever—use this theorem, so we won’t prove it here. For a proof andfurther discussion see e.g. [Hungerford]. (We will, however, prove and use the analogoustheorem for modules over a ring.)

10.3 Perfect groups

A group G is perfect if G = [G,G]. Thus G is perfect if and only if Gab = {e} if and only ifevery homomorphism from G to an abelian group is trivial. Some easy facts about perfectgroups:

• A group is both perfect and solvable if and only if it is trivial.

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• Any simple non-abelian group is perfect.

• Perfect groups are closed under quotients, extensions, and arbitrary products (but notunder subgroups).

Note that a perfect group need not be simple. For example, any product of non-abeliansimple groups such as A5 × A5 is perfect, but certainly not simple.

We will see below that with two exceptions, SLnF is perfect (but typically has non-trivialcenter, hence need not be simple). Indeed for us this is the main reason to bother introducingthe concept of a perfect group; we will use it as a stepping stone toward proving PSLnF is(almost always) simple.

11 On SLnF and PSLnF

Recall (see “Preliminary Remarks”) that SLnF is the group of n× n matrices over the fieldF with determinant 1. Note that GLnF fits into a split extension

SLnF−→GLnF−→F×.Many basic subgroups of SLnF are defined by simply intersecting with the corresponding

subgroups of GLnF . For example, define the Borel subgroup of SLnF by SBnF = BnF ∩SLnF , the upper triangular matrices with determinant 1. Although one rapidly grows tiredof putting the “S” in SBnF and its cousins, we’ll do so for a little while. Thus SDnFdenotes diagonal matrices of determinant 1, SUnF = UnF (since the unipotent subgroupalready consists of determinant 1 matrices), and SNnF = NnF ∩SLnF . Here one can checkthat SNnF = NSLnFSDnF—provided, as usual, that F 6= F2.

The one significant difference to watch out for concerns the Weyl group. One might thinkthat we should just take permutation matrices of determinant 1, but this turns out to bethe wrong thing to do. The way to think of it is as follows: In GLnF we have the extension

DnF−→NnF−→WnF,

where WnF ∼= Sn and a splitting of the extension is already given by using permutationmatrices. If F 6= F2 there is an analogous extension

SDnF−→SNnF−→SWnF,

where SWnF is by definition SNnF/SDnF and is again isomorphic to the symmetric groupSn. But unless char F = 2, the extension doesn’t split: the problem is that transpositions(as permutation matrices) have determinant −1 and there is no way to lift them to elementsof order 2 in SNnF . It’s already a problem when n = 2, as you can check. When char F 6= 2,they do however lift to elements of order 4; e.g. for n = 2 use(

0 −11 0

)

or its inverse. The upshot of this discussion is that we just have to live with the fact thatthe extension SDnF−→SNnF−→SWnF = Sn usually doesn’t split. In practice, fortunately,

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this does not cause any difficulties. To illustrate, let’s first clarify our definition of SWnF : Ifchar F = 2, it is just the permutation matrices as before, i.e. is equal to WnF . If char F 6= 2,SWnF = SNnF/SDnF . In all cases SWnF ∼= WnF ∼= Sn. Now consider the SLn version ofthe Bruhat decomposition.

Theorem 11.1SLnF =

∐w∈WnF

SBnFw̃SBnF,

where in the case char F 6= 2, w̃ is any preimage of w in SNnF (in the characteristic 2 caseit is just a permutation matrix as usual).

Proof/Discussion: First of all, note that the expression on the right is well-defined. Anytwo choices of w̃ differ by an element of SDnF ⊂ SBnF , so the double coset SBnFw̃SBnFis independent of the choice. The theorem then follows easily from its GLn analogue; nonew fussing about with row reduction is needed. Certainly the indicated double cosets aredistinct and hence disjoint, by comparing with the GLn result. And if g ∈ SLnF , we canwrite g = b1w̃b2 with b1, b2 ∈ BnF by the GLn result, with det b2 = (det b1)

−1. We can writeb1 = b′1d with b′1 ∈ SBnF and d ∈ DnF . Then since w̃ normalizes DnF , we have

g = b′1w̃(d′b2),

where det d′ = det d = det b1 and hence d′b2 ∈ SBnF .

Carrying around the decorations S, n, F gets tiresome, as does the w̃ notation. So aslong as we have firmly declared that our context is SLnF , we prefer to abbreviate the aboveBruhat decomposition in a slightly abusive but much more pleasant notation as

SLnF =∐w∈W

BwB.

Thus it is understood that B means SBnF , and so on.

11.1 On SL2F

The goal of this section is to study the structure of SL2F in detail, and in particular toprove:

Theorem 11.2 If F is a field, then PSL2F is a simple group except when F = F2 orF = F3.

In the exercises you show that PSL2F2 ∼= S3 and PSL2F3 ∼= A4, so these two cases arenot simple. Taking F = Fq to be a finite field with q > 3, the theorem gives us a large,potentially new class of simple groups. I say “potentially new” because it might happenthat some of these groups are isomorphic to alternating groups. Indeed |PSL2F5| = 60, andhence PSL2F5 ∼= A5 (see the exercise on simple groups of order 60). It turns out that thisis the only such coincidence, however, a fact that one can check by hand for small q by just

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checking the orders are not of the form n!/2. For example, PSL2F7 is simple of order 168and so is not an alternating group.

The group SL2F is important for much more general reasons, as it is the most basicexample of a “semi-simple algebraic group”, or in the case F = R,C of a “semi-simple Liegroup”. It plays a central role in representation theory and Lie theory, as well as in certainparts of algebraic topology, algebraic geometry, and combinatorics.

11.1.1 Some handy formulas in SL2

We will need some elementary formulas in SL2F . They are all easy to check directly. Firstsome notation: If t ∈ F we set x(t) = (

1 t0 1

)

and y(t) = (1 0t 1

)

If t ∈ F× we set w(t) = (0 t−t−1 0

)

and h(t) = (t 00 t−1

)

These satisfy:

1. w(1)x(t)w(1)−1 = y(−t)

2. h(t) = w(t)w(1)−1

3. w(t) = x(t)y(−t−1)x(t)

4. [h(s), x(t)] = x((s2 − 1)t)

5. h(s)x(t)h(s)−1 = x(s2t)

6. x(t)h(s)x(t)−1 = h(s−1)x((s2 − 1)t)

Note that any of the last three formulas easily determines the other two, and that thereare analogues with x(t) replaced by y(t).

Let w = w(1).

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11.1.2 Properties of SL2F

Note that the center C := C(SL2F ) is trivial if char F = 2 and is C2 (i.e. ±Id) otherwise.The Bruhat decomposition SL2F = B

∐BwB immediately implies:

Proposition 11.3 B is a maximal proper subgroup of SL2F .

Proposition 11.4 SL2F is generated by the subgroups U,U− (together), i.e. by the elementsx(t), y(t), t ∈ F .

Proof: Let K denote the subgroup generated by U,U−. By the Bruhat decomposition weknow that SL2F is generated by U , D, and w; that is, by the elements x(t), h(t), w(1). Sincew(1) ∈ K by Formula (3), and hence h(t) ∈ K by Formula (2), we are done.

Proposition 11.5 SL2F is perfect if F 6= F2, F3.

Proof: By the preceeding proposition, it suffices to show that x(a), y(a) are commutatorsfor all a ∈ F . By Formula (1), we need only consider x(a). By Formula (4) we see that x(a)is a commutator provided there is an s ∈ F× with s2 6= 1, and this latter statement is trueif and only if |F | > 3.

The two exceptional cases are not perfect (see the exercises).

11.1.3 Simplicity of PSL2

We restate the theorem:

Theorem 11.6 If F is a field, then PSL2F is a simple group except when F = F2 orF = F3.

Proof: Assume |F | > 3. We abbreviate G := SL2F . It is equivalent to show that if H is anormal subgroup of G, then either H ⊂ C (the center) or H = G. Since H is normal, HBis a subgroup of G, and as it contains B, by maximality of B we have either HB = B orHB = G.

Suppose HB = B. Then H ⊂ B, and as H is normal and wBw−1 = B−, we haveH ⊂ B ∩B− = D. But by Formula (6), any subgroup of D that is normal in G must consistof elements h(s) with s2 = 1, i.e. must lie in C.

Now suppose HB = G. Then G/H ∼= B/(B ∩ H). Since perfect groups and solvablegroups are preserved under quotients, we conclude using Proposition 11.5 (and the assump-tion |F | > 3) that G/H is both perfect and solvable, hence trivial. So H = G, as desired.

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11.2 Simplicity of PSLnF : a sketch

In this section we sketch how the results on SL2 extend to SLn. In particular we sketch aproof that PSLnF is simple for all n ≥ 3 and all F .

We use the following notation: For each i, 1 ≤ i < n, we let Gi ⊂ GLnF denote theblock diagonal subgroup consisting of just one 2× 2 SL2F block in the i, i+ 1 position. Forexample, if n = 4 then G2 is the subgroup

1 0 0 00 a b 00 c d 00 0 0 1

with ad − bc = 1. In each such block we have copies of the elements x(t), y(t), h(t), w(t)defined above for SL2F , which we denote xi(t), yi(t), etc. Similarly we have subgroupsUi, Bi, Di, Ni etc. of Gi corresponding to U,B,D,N in SL2F . Caution: This conflicts withour earlier notation,in which for example Bi denoted the upper triangular matrices in GLiF .In the displayed matrix above, B2 is the subgroup defined by c = 0.

Proposition 11.7 SLnF is generated by the subgroups Gi, and hence is generated by theelements xi(t), yi(t), t ∈ F .

Proof: The first statement follows by the usual row/column reduction (or Bruhat decomposi-tion) argument, since all the elementary row/column operations can be realized by repeatedleft/right multiplication by elements of the Gi’s. The second statement then follows by whatwe proved for SL2.

Proposition 11.8 SLnF is perfect except when n = 2, F = F2, F3.

Proof: The case n = 2 was proved earlier. If F 6= F2, F3, the general case follows immediatelyfrom the fact that the Gi’s are perfect together with the previous proposition. However, onecan give a uniform proof for all F and n ≥ 3 as follows: It suffices to show each xi(t) is acommutator. These are all conjugate in SLnF (via a permutation of coordinates), so we canassume i = 1, in which case it is enough to prove the result for n = 3. But x1(t) is conjugateto x13(t), i.e. 1 0 t

0 1 00 0 1

Since [x1(a), x2(b)] = x13(ab), x13(t) is a commutator for all t and we’re done.

Now the other key ingredient in the PSL2 simplicity proof was that the Borel subgroupB ⊂ PSL2F is a maximal subgroup. This is clearly not true for higher n, as there arevarious block-triangular groups that contain B. Explicitly if a = (a1, ..., ar) is an orderedpartition of n, there is an associated parabolic subgroup Pa consisting of block triangular

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matrices whose diagonal blocks have size a1, ..., ar. For example if n = 6 and a = (3, 1, 2),then Pa consists of matrices

a b c ∗ ∗ ∗d e f ∗ ∗ ∗g h i ∗ ∗ ∗0 0 0 j ∗ ∗0 0 0 0 k l0 0 0 0 p q

where the ∗ entries are arbitrary and the three diagonal blocks have the product of theirdeterminants equal to 1. Note that B itself corresponds to the partition (1, 1, ..., 1), whileSLnF corresponds to (n). Note also that Pa is generated by B together with the wi’s itcontains. The total number of parabolic subgroups is thus 2n−1. The surprising fact is:

Proposition 11.9 Suppose H is a subgroup of SLnF containing B. Then H = Pa for somea.

In particular there are only finitely many subgroups containing B. Note that when n = 2the proposition says that B is a maximal proper subgroup.

Theorem 11.10 PSLnF is simple unless n = 2 and F = F2, F3.

Proof: The case n = 2 was proved earlier, so assume n ≥ 3 and let H ⊂ SLnF be a normalsubgroup. Then HB is a subgroup containing B, so HB = Pa for some partition a. IfPa = B then H ⊂ B, and as in the case n = 2 we find that H ⊂ D, from which it followsthat in fact H ⊂ C, C being the center (i.e. scalar matrices of determinant 1). The readercan fill in the details of this step. If Pa = SLnF , then as in the case n = 2 we find thatSLnF/H is both perfect and solvable, hence trivial, so H = SLnF and we’re done.

The new step that remains is to show that the case Pa 6= B, SLnF can’t occur. Wesketch briefly the ideas involved. Suppose Pa 6= B, SLnF . Then Pa contains some butnot all of the wi’s. Hence there must exist i, i + 1 such that wi ∈ Pa and wi+1 /∈ Pa.Since BwiB ⊂ Pa = HB, we have (BwiB) ∩ H 6= ∅. Since H is normal, it follows that(wi+1BwiBw

−1i+1)∩H 6= ∅. From this one can show that wi+1wiwi+1 ∈ HB = Pa. But this is

false, as one can check from the definition of Pa. So we have a contradiction and the proofis complete.

Taking F to be a finite field, this yields a large family of finite simple groups. With avery small number of exceptions, they are distinct from each other and from the alternatinggroups (up to isomorphism). In fact if memory serves, the only (?) exception besides thecoincidence PSL2F5 ∼= A5 already mentioned is PSL2F7 ∼= GL3F2. Note these last twogroups have order 168. To get the isomorphism one can show that in fact there is only onesimple group of order 168 up to isomorphism; for an elaborate proof of this see [Dummit-Foote].

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12 Exercises

Note: Selected exercises will be assigned. Remember too that your mission is not merelyto find any old proof; always strive for a simple, elegant argument, and of course make fulluse of the machinery that we develop. Needless to say, a “simple, elegant” argument may ormay not come to mind, or even be possible, but it should always be your goal.

Notice. The Feit-Thompson theorem is off-limits unless explicitly allowed!

A. G-actions.

A1. By Cayley’s theorem (which is trivial, from a modern perspective), every finite groupis isomorphic to a subgroup of Sn for some n, where the n provided by the proof is n = |G|.On the other hand, G may well be isomorphic to a subgroup of Sm for some much smallerm (think of Sm itself, for example). Show, however, that for G = Q8 (the quaternion groupof order 8), n = 8 is the minimal n for which the conclusion of Cayley’s theorem holds.

A2. Prove by first finding a suitable set on which the group in question acts:a) GL2F2 is isomorphic to S3, and Aff2F2 is isomorphic to S4.b) PSL2F3 is isomorphic to A4.c) If G is a simple group of order 60, then G is isomorphic to A5.

B. Wreath products.

Let H be a group, G ⊂ Sn a subgroup. The wreath product H∫G is the semi-direct

product HnoG, where G acts on the left of Hn by permuting the factors. Thus the elementsof H

∫G have the form (h1, ..., hn)g, with multiplication determined by the formula

g · (h1, ..., hn) = (hg−11, ..., hg−1n)g.

Note that the notation is defective, since H∫G depends not just on G but on the particular

n and the way G is embedded in Sn. But this should cause no confusion in context. Inparticular the notation H

∫Sn always means Hn o Sn unless otherwise specified.

B1. Partition [mn] into n blocks (subsets) of equal size m, compatibly with the standardorder on [mn]. Let Γ ⊂ Smn denote the subgroup of all block-preserving permutations; bythis we mean that γ ∈ Γ is allowed to permute the blocks as well as the elements within aparticular block. Then Γ ∼= Sm

∫Sn.

B2. Suppose G ⊂ Sm and K ⊂ Sn. Regard Sm∫Sn as a subgroup of Smn as in 2.1. Then

there is an isomorphism

(H∫G)

∫K ∼= H

∫(G

∫K).

Consequently expressions such as H∫G1

∫G2...

∫Gr are unambiguous (assuming Gi is

given as a subgroup of Sni). In particular we can define the r-fold iterated wreath product∫ rH of a subgroup H ⊂ Sn. It is a subgroup of Snr .

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B3. Wreath products at Wimbledon. The Wimbledon tennis tournament begins with a“draw” of 128 players. Player no. 1 plays no. 2, no. 3 plays no. 4, the winner of 1-2 playsthe winner of 3-4, and so on. Some care is needed in determining the draw; for instance thetwo (theoretically) best players should be placed in opposite halves, say one at position 1and the other at position 128, so that if they meet at all it will be in the finals. The iteratedwreath product

∫ 7C2 ⊂ S128 can be thought of as the group of permutations of the drawleaving it “essentially unchanged”, and the number of “essentially distinct” possible drawsis (128!)/2127. Explain.

B4. Sylow subgroups of symmetric groups. Let n = a0 + a1p + ... + ampm be the p-adic

expansion of n (so 0 ≤ ai < p). Then Sn has a p-Sylow subgroup P with

P ∼=m∏i=1

(∫ i

Cp)ai .

Here∫ iCp is the i-th iterated wreath product of Cp, and for each i we are taking the product

of ai copies of it, 1 ≤ i ≤ m.

C. Sylow subgroups of general linear groups of finite fields. In this exercise p is a prime andq = pd for some d, while Fq denotes a field with q elements. For any prime `, ν`n denotesthe exponent of ` in the prime factorization of n.

C1. Answer/show: a) |GLnFq| = q(n2)∏n

i=1(qi − 1)

b) The unipotent subgroup U is a p-Sylow subgroup.c) How many p-Sylow subgroups are there?d) What is the order of SLnFq? Order of PSLnFq?Note: Recall that F×p is a cyclic group. This is true for any finite field, so F×q is cyclic of

order q − 1. We haven’t proved the general case yet, but you can assume it if necessary inpart (d).

Note: Some of the remaining C problems may require a little knowledge of finite fieldsbeyond what has been discussed in class, and therefore may be postponed.

C2. Now let ` be a prime 6= p. Assume that ` divides q−1, and that if ` = 2 then 4|(q−1).Then DW = NGLnFqD contains an `-Sylow subgroup of GLnFq. Hence if ν`(q−1) = a, GLnFqhas an `-Sylow subgroup isomorphic to C`a

∫L, where L is an `-Sylow subgroup of Sn (cf.

3.1).Remarks: (i). If ` is odd and ` doesn’t divide q − 1, the `-Sylow subgroups are of a

similar nature but the details are more complicated. To pursue this point, think about theextension of Fq obtained by adjoining an `-th root of unity.

(ii). Why the restriction when ` = 2? One of my favorite mottos is the doubly nonsensical“4 is an odd prime”. The key, and elementary, number-theoretic fact behind this motto isthe following: Suppose ν`(x− 1) = a ≥ 1. Then ν`(x

n − 1) = a+ ν`n provided that either `is odd or a ≥ 2. When ` = 2 and a = 1 this fails, e.g. for x = 3, n = 2.

C3. Let a = ν2(q2 − 1).

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a) If q = 3mod 4, the 2-Sylow subgroups of GL2Fq are isomorphic to the semi-dihedralgroup SDa+1 of order 2a+1, defined for a ≥ 3 as follows: Let C2a , C2 have generators x, yrespectively, and let C2 act on C2a by y · x = x2

a−1. Then SDa+1 = C2a o C2.Suggestion: Identify F2q with Fq2 and consider the group of units of Fq2 together with the

Frobenius a 7→ aq.

b) If q = 3mod 4 and n = 2m, then GL2Fq∫Sm (embedded in GLnFq in the evident way)

contains a 2-Sylow subgroup P . Hence P ∼= SDa+1

∫Q, where Q is a 2-Sylow subgroup of

Sm. What happens for n odd?

c) If q is any odd prime power, the 2-Sylow subgroups of SL2Fq are isomorphic to thegeneralized quaternion group Qa of order 2a. (Qa is defined as follows for a ≥ 3: Write thequaternions as H = C⊕ Cj. Then Qa is the subgroup of H+ generated by j and the 2a−1-stroots of unity in C.)

D. Balanced products and induced G-sets.

D1. Let F be a field, and let Xn ⊂MnF denote the subset consisting of matrices with ndistinct eigenvalues, all of which lie in F . In this problem you will give two alternate ways ofthinking about Xn. Let Zn = Dn∩Xn, which we identify with the subset of (F×)n consistingof n-tuples with no repeated entries.

a) Let Yn denote the set of ordered n-tuples (L1, ..., Ln) of lines in F n such that∑Li = F n.

Then there is a “natural” bijection Zn ×Sn Yn∼= Xn. (Here “natural” is meant informally,

i.e. the definition of the bijection should flow naturally out of the given data.)

b) Note that GLnF acts on Xn by conjugation, and Zn is invariant under the restrictionof this action to NnF . Use the recognition principle for induced G-sets to show that thecanonical map GLnF ×NnF Zn−→Xn is a bijection.

Remark: For fans of topology, I note that for F = R,C, the set Xn is a subspace of MnF(with its usual topology). The other two spaces in (a), (b) have quotient topologies makingthe above bijections homeomorphisms.

D2. Let H be a subgroup of G and let X be a G-set. Then there is a natural isomorphismof G-sets G×H X ∼= (G/H)×X, where the target has the product G-action. In particular,the induced action g1 · (g, x) = (g1g, x) on G×X is isomorphic to to the product action.

E. Maximal and minimal subgroups of nilpotent and solvable groups.“Maximal subgroup” means “maximal proper subgroup”, while “minimal subgroup”

means “minimal non-trivial subgroup”. Here are four interesting facts to prove; G is al-ways a finite group.

E1. Suppose G is solvable. Thena) Every minimal normal subgroup is an elementary abelian p-group. (A finite abelian

p-group A is elementary if every element of A has order p, or equivalently A is a product ofof Cp’s.)

b) Every maximal subgroup has prime power index.

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E2. Suppose G is nilpotent. Then:a) Every minimal normal subgroup is central of prime order.b) Every maximal subgroup is normal and has prime index.

E3. The converse of 2b holds in the form: If G is a finite group such every maximalsubgroup is normal, then G is nilpotent. (Suggestion: Show that every p-Sylow subgroup isnormal.)

F. Properties of unipotent and Borel groups.

1. UnF is generated by the subgroups Ui,i+1 for 1 ≤ i < n.2. C(UnF ) = U1,n.3. [UnF,UnF ] is the subgroup of elements whose (i, i+ 1) entries are zero, 1 ≤ i < n.4. Determine the ascending central, descending central and commutator series for U4F .

(Or if feeling ambitious, do it for general n.)

Structure of BnF .1. If F 6= F2, [BnF,BnF ] = UnF . Hence for all fields the commutator series of BnF is

determined by that of UnF .2. Determine the upper and lower central series of BnF for F 6= F2. (They won’t get

far!)

M. Miscellaneous problems. These are mostly problems already suggested in the bodyof the notes.

M1. Show that if either n = pq with p, q prime, or n = pqr with p, q, r distinct primes,then every group of order n is solvable.

M2. Without using Burnside’s paqb theorem, show that every group of order < 60 issolvable.

M3. Show that every finite nilpotent group is supersolvable.

M4. Show that an abelian group admits a Jordan-Holder filtration if and only if it isfinite.

M5. Let P be a p-Sylow subgroup of Sp (p prime). Show that NSpP∼= Aff1Fp.

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