for physics students Péter Bántay Institute for ...elmfiz.elte.hu/~bantay/grouptheor/gr_intro.pdf · mathematics (Galois theory, di erential equations, geometry, etc.) physics (crystallograph,y

Group theory

for physics students

Péter Bántay

Institute for Theoretical Physics, ELTE

Introduction

Group theory is currently one of the most important mathematical dis-

ciplines, with manifold applications in

· mathematics (Galois theory, di�erential equations, geometry, etc.)

· physics (crystallography, solid state physics, high energy physics,

gauge theories, phase transitions, general relativity, etc.)

· chemistry

· esoteric sciences (like economics, linguistics, etc.)

Question: why groups?

Origins of group concept: symmetry and (co)homology.

Homology is a topological notion characterizing the �connectedness� of

a manifold through a sequence of groups.

Has important physical applications in the study of quantum systems

(Berry phase), gauge theories (instantons), general relativity (gravita-

tional singularities) and string theory.

Can be generalized to more general situations (homological algebra).

Symmetry: invariance under suitable transformations.

Examples: bilateral (mirror) symmetry of the human body (characteris-

tic of a very large class of animals, the Bilateralia) to be contrasted with

the �ve-fold rotational symmetry of echinoderms and the icosahedral

symmetry of certain micro-organisms.

Similar symmetry patterns show up in arti�cial (man made) objects like

buildings, furniture, decorations, etc.

Groups describe the algebra of symmetries!

Group theory provides techniques to convert qualitative information into

quantitative one.

Examples:

1. Counting of phenomenological constants

σij = Lijklukl Hooke's law

with uij and σij denoting the deformation and the stress tensor.

Onsager's symmetry relations

Lijkl = Ljikl = Lklij = · · ·

Question: number of independent elastic coe�cients Lijkl?

Answer: 2 in an isotropic medium (the elastic moduli), 21 in a triclinic

crystal.

For a crystal with point group G (in any dimension)

1

8

∑g∈G

{Tr(g)

4+ 2Tr(g)

2Tr(g2)

+ 3Tr(g2)2

+ 2Tr(g4)}

More generally, the free energy (a scalar quantity) is an invariant poly-

nomial of the (symmetric) deformation tensor, hence the answer equals

the number of independent invariants of the symmetrized square of the

point group (above formula gives the number of quadratic invariants,

appropriate in the linear case).

2. Reduction of partial di�erential equations

The solutions of a boundary value problem with given symmetry group

can only depend on the invariants of that group solution may be

reduced to a problem with fever independent variables, by introducing

some of the invariants as new variables (the solution will only depend on

these).

As an example, consider a spherically symmetric potential V (r). Then

Schrödinger's equation for the wave function ψ(r, φ, ϑ)

− ~2

2m∆ψ + V ψ = Eψ

(describing the motion of a point particle of mass m) is rotationally

invariant, hence in case of symmetric boundary conditions (that are in-

dependent of the angle coordinates r, e.g. ψ(R,φ, ϑ) =ψ0 for some R)

its solution is independent of φ and ϑ, and the problem is reduced to

solving the ODE

1

r2d

dr

(r2dψ

dr

)+ V (r)ψ = Eψ

with boundary condition ψ(R)=ψ0.

Remark. Usually, because the boundary conditions can break some

symmetries of the equation, not all of them play a role (unlike in the

previous example).

1 HISTORICAL HIGHLIGHTS

1 Historical highlights

Antiquity: application of symmetry principles in geometry (Euclid of

Alexandria, Archimedes of Syracuse), classi�cation of platonic solids.

J.-L. Lagrange (1771): solubility of polynomial equations.

É. Galois (1832): Galois theory.


A. Cayley (1854): abstract group concept.

É. Mathieu (1861,1873): Mathieu groups.

F. Klein (1873): classi�cation of geometries (Erlangenprogram).

S. Lie (1871-1893): continuous transformation groups.


H. Poincaré (1882): homology groups, uniformization.

É. Picard (1883): di�erential Galois theory.

D. Hilbert (1888): invariant theory, homological algebra.


W. Killing (1880-1890), É. Cartan (1894): classi�cation of simple Lie-

algebras.

G.F. Frobenius (1896): representation theory, group char-

acters.

W. Burnside (1903): �nite groups.


I. Schur (1904): projective representations.

A. Haar (1933): invariant integrals.

1966-1976: classi�cation of �nite simple groups.


Group theory in physics

Renaissance: symmetry principles in statics (�Epitaph of Stevinus�).

L. Euler (1765): movement of rigid bodies.

E.S. Fedorov (1891), L. Schön�ies (1891) és W. Barlow

(1894): classi�cation of crystal structures in 3D.

H. Poincaré (1900): symmetries of Maxwell's equations.


E. Noether (1915): symmetries and conservation laws.

E. Wigner: symmetries in quantum physics (1933),

relativistic wave equations (1947).

C.N. Yang és R. Mills (1954): gauge theories.

M. Gell-Mann (1963): �eightfold way� (leading to the quark model).

D. Shechtman (1982): quasi-crystals.

2 FUNDAMENTAL CONCEPTS

2 Fundamental concepts

Q: What is a group? How to compare groups?

A group is a set of elements with a suitable binary operation.

Binary operation: rule that assigns to two elements of a set a well-de�ned

third element of the same set.

Examples: addition and multiplication of (integer, rational, real, com-

plex, hypercomplex, p-adic, etc.) numbers, greatest common divisor and

lowest common multiple of integers, addition and cross product of vec-

tors (but not their scalar product), addition and multiplication of linear

operators, polynomials, matrices, ...


Multiplicative in�x notation: for a binary operation on the set X, we

denote by x?y (or simply by xy) the element obtained by applying the

operation to the elements x, y∈X, and call it their product.

A binary operation (on the set X) is called

associative, if for any x, y, z∈X

x(yz) = (xy)z

commutative, if for any x, y∈X

xy = yx

unital, if there exists 1X∈X (the identity) such that for all x∈X

1Xx = x1X = x


A group is a set G of elements together with an associative and unital

binary operation (the 'product'), such that for all x ∈ G there exists

x-1∈G (the inverse of x) for which

xx−1 = x−1x = 1G

A homomorphism is a mapping φ :G1→G2 such that for all x, y∈G1

φ(xy)=φ(x)φ(y)

An isomorphism is a bijective (i.e. one-to-one) homomorphism.

The groups G and H are isomorphic, denoted G∼=H, if there exists an

isomorphism φ :G→H.


Re�exive (G ∼= G), symmetric (G ∼= H implies H ∼= G) and transitive

(G∼=H and H∼=K implies G∼=K) relation.

Isomorphy principle: isomorphic groups cannot be distinguished alge-

braically.

Automorphism: isomorphism α : G→ G of a group with itself. Auto-

morphisms form a group, denoted Aut(G), with composition as binary

operation (symmetries of algebraic structure).

Order of a group: cardinality∣∣G∣∣ of its set of elements.

The orders of isomorphic groups coincide: G∼=H ∣∣G∣∣= ∣∣H∣∣.

Abelian group: group product is commutative (very special properties).


Generalizations of the group concept:

· relaxing the existence of inverses leads to monoids (with applica-

tions to automata theory & linguistics, renormalization, etc.);

· relaxing the associativity of the product results in quasi-groups

(combinatorial applications like latin squares, aka. sudoku);

· a partially de�ned product gives groupoids (with applications to

topology, the description of quasi-crystals, etc.);

· theoretical physics quantum groups, supergroups, ...


Recommended reading:

A.A. Kirillov : Elements of the theory of representations, Springer (1976).

J.L. Alperin and B. Bell : Groups and representations, Springer (1995).

D. Robinson : A course in the theory of groups, Springer (1993).

W. Magnus, A. Karrass and D. Solitar: Combinatorial group theory,

Interscience Publishers (1966).

3 EXAMPLES

3 Examples

3.1 Additive and unit groups of rings

Consider a (unital) ring R , i.e. a set with two associative and unital

binary operations, addition ('+') and multiplication ('·'), that satisfy

1. addition is commutative, and each a ∈R has an additive inverse,

denoted −a, such that a+(−a)=(−a)+a=0, where 0 denotes the

additive identity (the zero element of R);

2. the distributive laws a (b+c)=ab+ac and (a+b) c=ac+bc hold for

any a, b, c∈R.

3 EXAMPLES

Examples of rings:

1. The sets Z, Q, R, C of integers, rationals, reals, resp. complex

numbers, with the usual addition and multiplication are commu-

tative rings (but the set 2Z+1 of odd integers is not a ring, since

the sum of two odd numbers is even).

2. The set Z/nZ = {nZ+k | 0≤k

3 EXAMPLES

3. For a commutative ringR and a setX, the set ofR-valued functions

on X (i.e. maps from X to R) with pointwise operations

(f + g)(x) = f(x) + g(x)

(fg)(x) = f(x) g(x)

for x∈X, is again a commutative ring, denoted usually R(X). If

both R and X have some kind of geometric structure (e.g. topo-

logical, di�erentiable, complex analytic, etc.), then the structure

preserving (i.e. continuous, di�erentiable, holomorphic, etc.) maps

form a ring themselves.

3 EXAMPLES

4. For a commutative ring R and a positive integer n, a polynomial

in n indeterminates is a map from Zn+ into R with �nite support,

i.e. which takes on non-zero values at only a �nite number of

arguments. The addition of polynomials is de�ned pointwise, while

the product of the polynomials P and Q reads

(PQ)(α1, . . . , αn) =∑

0≤βi≤αi

P (β1, . . . , βn)Q(α1−β1, . . . , αn−βn)

With these operations, polynomials form the ring R[x1, . . . , xn].

The indeterminates (generating the whole ring) are de�ned as the

polynomials xi(α1, . . . , αn)=δαi,1∏j 6=i

δαj ,0.

3 EXAMPLES

5. For two positive integers n and m and a commutative ring R, the

maps from the set X = {1, . . . , n}×{1, . . . ,m} into R are called

n-by-m matrices over R, and their set is denoted Matn,m(R); a

matrix is square of size n if m = n. The values of the matrix

A ∈ Matn,m(R) are called its matrix elements, and are denoted

Aij = A(i, j) for 1 ≤ i ≤ n and 1 ≤ j ≤ m. The addition of the

matrices A ∈Matn,m(R) and B ∈Matp,q(R) is de�ned pointwise,

(A+B)ij=Aij+Bij , provided p=n and q=m, while their product

is de�ned when p=m, in which case it reads (AB)ij=m∑k=1

AikBkj .

It follows that the set Matn(R) of square matrices of any size n

forms a ring, which is not commutative unless n=1.

3 EXAMPLES

The set of elements of a ring R with the operation of addition forms an

Abelian group, the additive group of R.

A unit is an element a∈R having a multiplicative inverse a−1 such that

aa−1 = a−1a = 1R

denoting by 1R the multiplicative identity ('one') of R.

The set of units with the operation of multiplication forms a group R×,

the unit group of R.

A �eld F (like Q, R or C) is a ring whose unit group contains all its

nonzero elements: F×=F \{0}.

3 EXAMPLES

Examples:

1. Z×={±1};

2. (Z/nZ)× = {nZ+k | 0≤k0.

5. Matn(R)×

= GLn(R) = {A∈Matn(R) | detA∈R×}, the general

linear group over R, consisting of the invertible n-by-n matrices.

3 EXAMPLES

3.2 Matrix groups

Besides the general linear group GLn(R), several other subsets of Matn(R)

form a group with the operation of matrix multiplication:

· the set∆n(R) = {A∈GLn(R) |Aij=0 if i 6=j}

of diagonal matrices is an Abelian group;

· the setMn(R) of monomial matrices, in which each row and column

contains exactly one nonzero entry;

3 EXAMPLES

· the set

Πn = {A∈Mn(R) |Aij 6=0 implies Aij=1R}

of permutation matrices (note that every monomial matrix is the

product of a diagonal and a permutation matrix);

· the setOn(R) =

{A∈GLn(R) |A−1 =A

ᵀ}of orthogonal matrices (where A

ᵀdenotes the transpose of a ma-

trix A∈Matn(R), with entries (Aᵀ)ij = Aji), and more generally

Op,q(R) ={A∈GLp+q(R) |A−1 =η−1p,qA

ᵀηp,q

}with ηp,q denoting the diagonal matrix having the �rst p diagonal

entries equal to 1R, and the remaining q entries equal to −1R;

3 EXAMPLES

· the set

Sp2n(R) ={A∈GL2n(R) |A−1 =J−1n A

ᵀJn

}of symplectic matrices, where Jn is the block-diagonal matrix made

up of n copies of the Pauli-matrix

iσ2 =

(0 1R

−1R 0

)

· subsets of all the above whose elements satisfy the requirementdetA=1R, e.g. the special linear and orthogonal groups

SLn(R) = {A∈GLn(R) | detA=1R}

SOn(R) = {A∈On(R) | detA=1R}

3 EXAMPLES

3.3 Symmetric and alternating groups

Permutation: bijective self-map of a (�nite) set into itself.

Product of permutations: composition of bijective maps (associative op-

eration, with the trivial permutation idX:x 7→x as identity element).

Inverse of a permutation: inverse map (again bijective).

The collection of all permutations of a �nite setX forms a group Sym(X),

the symmetric (NOT symmetry) group over X (not commutative in case

|X|>2), of order ∣∣Sym(X)∣∣ = ∣∣X∣∣!Transposition: interchange of two elements.

3 EXAMPLES

Any permutation can be decomposed (in many ways) into a product of

transpositions: while their number is not unique, their parity (whether

there is an even or odd number) characterizes the permutation odd

and even permutations.

The product of even permutations is even even permutations form

themselves a group, the alternating group Alt(X) (of order half of that

of Sym(X)).

Sym(X) and Sym(Y ) (resp. Alt(X) and Alt(Y )) are isomorphic exactly

when∣∣X∣∣= ∣∣Y ∣∣

Sn=Sym({1, . . . , n}) and An=Alt({1, . . . , n}), the symmetric (alternat-

ing) groups of degree n (isomorphic to Πn and SΠn respectively).

3 EXAMPLES

3.4 Geometric symmetry groups

Symmetries of geometric �gure: rigid motions (Euclidean isometries)

mapping the �gure (as a set of points) onto itself.

Regular polygon: convex plane �gure all of whose sides are congruent

(i.e. have equal length) angles between neighboring sides are also

equal.

Up to similarity, exactly one regular n-gon for each integer n>2.

3 EXAMPLES

Medians (edge bisectors) of a regular polygon all meet in a single point,

the center of the polygon.

Symmetries of a regular n-gon form the dihedral group Dn of degree n

composed of rotations around the center (by multiples of 2π/n) & re�ec-

tions across lines passing through the center and some vertex.

3

σ_1

C

1

σ_2

σ_3

2

3 EXAMPLES

Since there are n di�erent re�ection axes (the medians for odd n, and for

even n the medians together with the diagonals passing through opposite

vertices) and n di�erent rotations, the order of the dihedral group is∣∣Dn∣∣=2n.Group structure can be described (for �nite groups) using the Cayley

(multiplication) table.

1G . . . h . . .

1G 1G . . . h . . .

......

. . ....

. . .

g g . . . gh . . .

......

. . ....

. . .

3 EXAMPLES

Symmetries map the polygon onto itself any set of distinguished sub-

�gures (like vertices, edges, medians, etc.) is also mapped onto itself.

Consequence: to each symmetry transformation corresponds a per-

mutation of the set X of distinguished sub�gures, hence there exists a

homomorphism πX : Dn → Sym(X) into the symmetric group over X

(can be handy for the computation of the multiplication table).

Since the set V of vertices of a regular n-gon has cardinality n, the

homomorphism πV is bijective only for n=3, when

∣∣Sym(V )∣∣=n!=2n= ∣∣Dn∣∣

3 EXAMPLES

1 C C2 σ1 σ2 σ3 πV

1 1 C C2 σ1 σ2 σ3 ()

C C C2 1 σ3 σ1 σ2 (1, 2, 3)

C2 C2 1 C σ2 σ3 σ1 (1, 3, 2)

σ1 σ1 σ2 σ3 1 C C2 (2, 3)

σ2 σ2 σ3 σ1 C2 1 C (1, 3)

σ3 σ3 σ1 σ2 C C2 1 (1, 2)

Multiplication table of D3 and the homomorphism πV : D3→S3.

3 EXAMPLES

Platonic solid (regular polyhedron): convex spatial �gure all of whose

bounding facets are congruent regular polygons.

Five Platonic solids (up to similarity):

1. tetrahedron (4 triangles);

2. octahedron (8 triangles);

3. icosahedron (20 triangles);

4. cube (6 squares);

5. dodecahedron (12 pentagons).

3 EXAMPLES

3 EXAMPLES

Symmetry groups of regular polyhedra:

· T ∼= A4 (tetrahedron),

· O ∼= S4 (octahedron cube)

· I ∼= A5 (icosahedron dodecahedron).

Regular convex polytopes in 4D: simplex (5-cell), orthoplex (16-cell),

hypercube (8-cell), 600-cell, 120-cell, 24-cell.

Only 3 regular polytopes in dimensions >4: the simplex, the orthoplex

(cross-polytope) and the hypercube.

3 EXAMPLES

3.5 Molecular symmetry groups

Charge density in molecules invariant under a �nite group (point group)

of geometric transformations, leading to restrictions on

1. the structure of the molecular spectrum (Wigner, Tisza);

2. electromagnetic characteristics (dipole and magnetic moments;

3. chemical properties.

3 EXAMPLES

Point groups (in 3D): 7 in�nite families + 7 polyhedral groups

1. Cn∼=Zn (cyclic), e.g. hydrogen peroxide H2O2 (n=2);

2. Cnv∼=Dn (pyramidal), e.g. water (n=2), ammonia (n=3), hydro-

gen �uoride (n=∞);

3. Cnh∼=Zn×Z2 , e.g. boric acid (n=3);

4. Dn∼=Dn (dihedral), e.g. twistane C10H16 (n=2);

5. Dnv ∼= D2n (anti-prismatic), e.g. ethane C2H6 (n = 3), ferrocene

Fe (C5H5)2 (n=5);

6. Dnh∼=Dn×Z2 (prismatic), e.g. ethylene C2H4 (n=2), boron tri�u-

oride BF3 (n=3), fullerene C70 (n=5), carbon dioxide CO2 (n=∞);

3 EXAMPLES

7. Sn∼=Z2n;

8. T∼=A4 (chiral tetrahedral);

9. Td ∼=S4 (tetrahedral), e.g. methane CH4;

10. Th∼=A4×Z2 (pyritohedral);

11. O∼=S4 (chiral octahedral);

12. Oh∼=S4×Z2 (octahedral), e.g. sulfur hexa�uoride SF6;

13. I∼=A5 (chiral icosahedral);

14. Ih∼=A5×Z2 (icosahedral), e.g. C60 fullerene.

3 EXAMPLES

3.6 Crystalline symmetry groups

Some homogeneous substances exhibit anisotropic behavior on a macro-

scopic scale (e.g. mechanical, optical, electric, etc. properties of crys-

tals).

At the phenomenological level: material characteristics like permittivity,

conductivity, etc. are tensorial (rather than scalar) quantities.

Since both microscopic homogeneity, both unordered microscopic inho-

mogeneity leads to isotropic behavior, anisotropy is a consequence of

ordered microscopic inhomogeneity.

Macroscopic order from discrete translational symmetries.

3 EXAMPLES

Crystalline substance: microscopic components (atoms, molecules or

ions) distributed periodically in space.

Quasi-crystal: aperiodic structure exhibiting long range order.

3 EXAMPLES

In a crystal, the microscopic components are localized (in the absence

of defects) around the lattice points of a 3D periodic lattice (the crystal

lattice).

Space group: transformations mapping the crystal lattice into itself

(composed of translations, rotations, re�ections, inversions and various

combinations of the above).

3 EXAMPLES

Translation subgroup: group of pure translations taking the crystal lat-

tice into itself.

Point group: �nite group describing the rotational and re�exion symme-

tries of a crystal.

Crystal structures grouped into crystal classes, families and systems ac-

cording to their point groups and translation subgroups.

3 EXAMPLES

Order of transformation: smallest positive integer N such that the Nth

power of the transformation is the identity.

Crystallographic restriction: the number of integers coprime to the order

of any element of the point group cannot exceed the dimension (valid only

for periodic structures, not quasi-crystals).

dimension 2, 3 4, 5 6, 7

allowed values {2, 3, 4, 6} ∪{5, 8, 10, 12} ∪{7, 9, 14, 18}

Consequence: only �nitely many di�erent crystal structures in any di-

mension.

3 EXAMPLES

Classi�cation results

dimension 2 3† 4‡ 5∗ 6∗

# crystal systems 4 7 33 59 251

# point groups 10 32 227 955 7104

# space groups 17 230 4894 222097 28934974

†: Fedorov (1891), Schön�ies (1891) and Barlow (1894).

‡: Brown, Bülow and Neubüser (1978).

∗: Plesken and Schulz (2000).

3 EXAMPLES

3.7 Space-time symmetry groups

Kinematics: description of the movement of material bodies, i.e. of the

time evolution of their mutual emplacements (distances).

Frame of reference: system of bodies with known relative motions.

The relative motions of all bodies of the Universe are completely deter-

mined by their motions with respect to a speci�c frame.

Question: which frames are the most useful?

Answer: inertial frames, in which the inertial motion of isolated bodies

(not interacting with the rest of the Universe) is uniform translation

( inertial frames move at constant speed relative to each other).

3 EXAMPLES

Galileo's relativity principle: all mechanical laws of motion are the same

in all inertial frames.

Newtonian mechanics: action at a distance (forces act instantly, without

any delay) & physical space is 3D Euclidean and time is a 1D continuum.

Basic symmetries of classical mechanics (in inertial frames):

· both space and time are homogeneous (no preferred origin);

· space is isotropic (no preferred directions);

· any reference frame obtained via a boost (constant speed uniform

translation) is again inertial.

3 EXAMPLES

Galilei group G consisting of

· space translations (3 parameters)

· time translations (1 parameter)

· spatial rotations (3 parameters)

· (Galilean) boosts (3 parameters)

· discrete re�ection symmetries

Noether's theorem: to each one-parameter group of continuous

symmetries of a physical system corresponds a conserved quantity.

3 EXAMPLES

Galilean symmetries correspond to universal �rst integrals.

space translations (linear) momentum

spatial rotations angular momentum

time translations energy

(Galilean) boosts center of mass

Poincaré: the symmetry group of Maxwell's equations (the so-called

Poincaré group P) is the isometry group of 4D (homogeneous and isotropic)

Minkowski space, not the Galilei group.

The Galilei group may be obtained by a limiting procedure (Wigner-

Inönü contraction) from the Poincaré group.

3 EXAMPLES

Einstein: Galileo's relativity principle holds for all physical laws (includ-

ing those of electrodynamics), and there is a limit speed (the speed of

light c) for the propagation of physical causes (no action at a distance)

the true symmetry group of physical laws (at arbitrary speeds,

but neglecting the e�ects of gravity) is the Poincaré group P.

The Poincaré group P consists of

· space-time translations (4 parameters)

· space-time rotations, including the spatial rotations and the Lorentz-

boosts (6 parameters)

· 4D re�ections

3 EXAMPLES

Two relativistic �rst integrals: a 4D vector (four-momentum) and a 4D

antisymmetric tensor (angular momentum).

Pauli-Lubanski (pseudo)vector: four-vector combining 4D linear and

angular momentum, describing helicity (spin) states of moving particles.

General relativity: �at Minkowski space-time has to be replaced by a

suitable curved manifold (a solution of Einstein's equations) Poincaré

group has to be replaced by the corresponding isometry group (e.g. the

de Sitter group).

Historical highlightsFundamental conceptsExamples

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for physics students Péter Bántay Institute for ...elmfiz.elte.hu/~bantay/grouptheor/gr_intro.pdf · mathematics (Galois theory, di erential equations, geometry, etc.) physics (crystallograph,y