Alperin weights Groups of odd order

Normal subgroups and a charactercorrespondence in groups of odd order

J.P. CosseyUniversity of Akron

Conference on Character Theory of Finite Groupsin honor of I.M. Isaacs

June 5, 2009

J.P. Cossey University of Akron Conference on Character Theory of Finite Groups in honor of I.M. Isaacs

Normal subgroups and a character correspondence in groups of odd order

Alperin weights Groups of odd order

Current status

Weights

Recall that if G is a finite group and p is a prime, a p-weight of Gis a pair (Q, ψ), where Q is a p-subgroup of G andψ ∈ Irr(NG (Q)/Q) is such that ψ(1)p = |NG (Q) : Q|p.

Of course, G acts by conjugation on the set of weights of G , andthe Alperin weight conjecture is the following:

Conjecture Suppose G is a finite group, and let p be a fixedprime. Then the number of irreducible Brauer characters of G isequal to the number of conjugacy classes of p-weights of G .

J.P. Cossey University of Akron Conference on Character Theory of Finite Groups in honor of I.M. Isaacs

Normal subgroups and a character correspondence in groups of odd order

Alperin weights Groups of odd order

Current status

Weights

Recall that if G is a finite group and p is a prime, a p-weight of Gis a pair (Q, ψ), where Q is a p-subgroup of G andψ ∈ Irr(NG (Q)/Q) is such that ψ(1)p = |NG (Q) : Q|p.

Of course, G acts by conjugation on the set of weights of G , andthe Alperin weight conjecture is the following:

Conjecture Suppose G is a finite group, and let p be a fixedprime. Then the number of irreducible Brauer characters of G isequal to the number of conjugacy classes of p-weights of G .

J.P. Cossey University of Akron Conference on Character Theory of Finite Groups in honor of I.M. Isaacs

Normal subgroups and a character correspondence in groups of odd order

Alperin weights Groups of odd order

Current status

Weights

Recall that if G is a finite group and p is a prime, a p-weight of Gis a pair (Q, ψ), where Q is a p-subgroup of G andψ ∈ Irr(NG (Q)/Q) is such that ψ(1)p = |NG (Q) : Q|p.

Of course, G acts by conjugation on the set of weights of G , andthe Alperin weight conjecture is the following:

Conjecture Suppose G is a finite group, and let p be a fixedprime. Then the number of irreducible Brauer characters of G isequal to the number of conjugacy classes of p-weights of G .

J.P. Cossey University of Akron Conference on Character Theory of Finite Groups in honor of I.M. Isaacs

Normal subgroups and a character correspondence in groups of odd order

Alperin weights Groups of odd order

Current status

Known cases

The Alperin weight conjecture is known to be true in the followingcases:

The symmetric groups (Alperin, Fong)

General linear groups (Alperin, Fong)

Various sporadic simple groups

Solvable groups (Isaacs, Navarro)

For the most part, there is not a known explicit bijection betweenthe two sets.

J.P. Cossey University of Akron Conference on Character Theory of Finite Groups in honor of I.M. Isaacs

Normal subgroups and a character correspondence in groups of odd order

Alperin weights Groups of odd order

Current status

Known cases

The Alperin weight conjecture is known to be true in the followingcases:

The symmetric groups (Alperin, Fong)

General linear groups (Alperin, Fong)

Various sporadic simple groups

Solvable groups (Isaacs, Navarro)

For the most part, there is not a known explicit bijection betweenthe two sets.

J.P. Cossey University of Akron Conference on Character Theory of Finite Groups in honor of I.M. Isaacs

Normal subgroups and a character correspondence in groups of odd order

Alperin weights Groups of odd order

Current status

Known cases

The Alperin weight conjecture is known to be true in the followingcases:

The symmetric groups (Alperin, Fong)

General linear groups (Alperin, Fong)

Various sporadic simple groups

Solvable groups (Isaacs, Navarro)

For the most part, there is not a known explicit bijection betweenthe two sets.

J.P. Cossey University of Akron Conference on Character Theory of Finite Groups in honor of I.M. Isaacs

Normal subgroups and a character correspondence in groups of odd order

Alperin weights Groups of odd order

Current status

Known cases

The Alperin weight conjecture is known to be true in the followingcases:

The symmetric groups (Alperin, Fong)

General linear groups (Alperin, Fong)

Various sporadic simple groups

Solvable groups (Isaacs, Navarro)

For the most part, there is not a known explicit bijection betweenthe two sets.

J.P. Cossey University of Akron Conference on Character Theory of Finite Groups in honor of I.M. Isaacs

Normal subgroups and a character correspondence in groups of odd order

Alperin weights Groups of odd order

Current status

Known cases

The Alperin weight conjecture is known to be true in the followingcases:

The symmetric groups (Alperin, Fong)

General linear groups (Alperin, Fong)

Various sporadic simple groups

Solvable groups (Isaacs, Navarro)

For the most part, there is not a known explicit bijection betweenthe two sets.

J.P. Cossey University of Akron Conference on Character Theory of Finite Groups in honor of I.M. Isaacs

Normal subgroups and a character correspondence in groups of odd order

Alperin weights Groups of odd order

Current status

Known cases

The Alperin weight conjecture is known to be true in the followingcases:

The symmetric groups (Alperin, Fong)

General linear groups (Alperin, Fong)

Various sporadic simple groups

Solvable groups (Isaacs, Navarro)

For the most part, there is not a known explicit bijection betweenthe two sets.

J.P. Cossey University of Akron Conference on Character Theory of Finite Groups in honor of I.M. Isaacs

Normal subgroups and a character correspondence in groups of odd order

Alperin weights Groups of odd order

Solvable groups

Vertices of Brauer characters

Definition

Let G be a solvable group and let ϕ ∈ IBrp(G ). If there exists asubgroup H of G and a character α ∈ IBrp(H) of p′-degree suchthat αG = ϕ, then we say any Sylow p-subgroup Q of H is avertex of ϕ.

For solvable groups, this definition is equivalent to Green’sdefinition of a vertex subgroup. Isaacs and Navarro have shownthat for solvable groups, a vertex subgroup for ϕ always exists andis unique up to conjugacy.

J.P. Cossey University of Akron Conference on Character Theory of Finite Groups in honor of I.M. Isaacs

Normal subgroups and a character correspondence in groups of odd order

Alperin weights Groups of odd order

Solvable groups

Vertices of Brauer characters

Definition

Let G be a solvable group and let ϕ ∈ IBrp(G ). If there exists asubgroup H of G and a character α ∈ IBrp(H) of p′-degree suchthat αG = ϕ, then we say any Sylow p-subgroup Q of H is avertex of ϕ.

For solvable groups, this definition is equivalent to Green’sdefinition of a vertex subgroup. Isaacs and Navarro have shownthat for solvable groups, a vertex subgroup for ϕ always exists andis unique up to conjugacy.

J.P. Cossey University of Akron Conference on Character Theory of Finite Groups in honor of I.M. Isaacs

Normal subgroups and a character correspondence in groups of odd order

Alperin weights Groups of odd order

Solvable groups

A vertex version of the Alperin weight conjecture

Isaacs and Navarro proved the following:

Theorem

(Isaacs, Navarro 1993) Let G be solvable and let Q be ap-subgroup of G . Then the number of irreducible Brauercharacters of G with vertex Q is equal to the number of weightswith first component Q.

Summing over all of the conjugacy classes of p-subgroups of G , werecover the Alperin weight conjecture.

However, we still do not have an explicit bijection in this case.

J.P. Cossey University of Akron Conference on Character Theory of Finite Groups in honor of I.M. Isaacs

Normal subgroups and a character correspondence in groups of odd order

Alperin weights Groups of odd order

Solvable groups

A vertex version of the Alperin weight conjecture

Isaacs and Navarro proved the following:

Theorem

(Isaacs, Navarro 1993) Let G be solvable and let Q be ap-subgroup of G . Then the number of irreducible Brauercharacters of G with vertex Q is equal to the number of weightswith first component Q.

Summing over all of the conjugacy classes of p-subgroups of G , werecover the Alperin weight conjecture.

However, we still do not have an explicit bijection in this case.

J.P. Cossey University of Akron Conference on Character Theory of Finite Groups in honor of I.M. Isaacs

Normal subgroups and a character correspondence in groups of odd order

Alperin weights Groups of odd order

Solvable groups

A vertex version of the Alperin weight conjecture

Isaacs and Navarro proved the following:

Theorem

(Isaacs, Navarro 1993) Let G be solvable and let Q be ap-subgroup of G . Then the number of irreducible Brauercharacters of G with vertex Q is equal to the number of weightswith first component Q.

Summing over all of the conjugacy classes of p-subgroups of G , werecover the Alperin weight conjecture.

However, we still do not have an explicit bijection in this case.

J.P. Cossey University of Akron Conference on Character Theory of Finite Groups in honor of I.M. Isaacs

Normal subgroups and a character correspondence in groups of odd order

Alperin weights Groups of odd order

Normal subgroups

Navarro’s bijection

Notation: Let IBrp(G |Q) denote the set of Brauer characters of Gwith vertex subgroup Q, and let w(Q) denote the set of weights ofG with first component Q.

In the case that G has odd order, Navarro has constructed, usingIsaacs’ “brown paper”, an explicit bijection

ϕ→ ϕ̃

from IBrp(G |Q) to w(Q).

One can show relatively easily that w(Q) = IBrp(NG (Q)|Q) in thiscase.

J.P. Cossey University of Akron Conference on Character Theory of Finite Groups in honor of I.M. Isaacs

Normal subgroups and a character correspondence in groups of odd order

Alperin weights Groups of odd order

Normal subgroups

Navarro’s bijection

Notation: Let IBrp(G |Q) denote the set of Brauer characters of Gwith vertex subgroup Q, and let w(Q) denote the set of weights ofG with first component Q.

In the case that G has odd order, Navarro has constructed, usingIsaacs’ “brown paper”, an explicit bijection

ϕ→ ϕ̃

from IBrp(G |Q) to w(Q).

One can show relatively easily that w(Q) = IBrp(NG (Q)|Q) in thiscase.

J.P. Cossey University of Akron Conference on Character Theory of Finite Groups in honor of I.M. Isaacs

Normal subgroups and a character correspondence in groups of odd order

Alperin weights Groups of odd order

Normal subgroups

Navarro’s bijection

Notation: Let IBrp(G |Q) denote the set of Brauer characters of Gwith vertex subgroup Q, and let w(Q) denote the set of weights ofG with first component Q.

In the case that G has odd order, Navarro has constructed, usingIsaacs’ “brown paper”, an explicit bijection

ϕ→ ϕ̃

from IBrp(G |Q) to w(Q).

One can show relatively easily that w(Q) = IBrp(NG (Q)|Q) in thiscase.

J.P. Cossey University of Akron Conference on Character Theory of Finite Groups in honor of I.M. Isaacs

Normal subgroups and a character correspondence in groups of odd order

Alperin weights Groups of odd order

Normal subgroups

Where do normal subgroups fit in?

We now ask the question, how does Navarro’s map behave withrespect to normal subgroups of the odd order group G ?

Suppose N C G and that Q is a p-subgroup of G , and letP = Q ∩ N. Certainly P C Q and NG (Q) ⊆ NG (P).

Since the map ϕ→ ϕ̃ is a bijection, then there is a uniquecharacter ϕ1 ∈ IBrp(NG (P)|Q) such that ϕ̃1 = ϕ̃. In other words,the map ϕ→ ϕ1 is a bijection from IBrp(G |Q) to IBrp(NG (P)|Q).

J.P. Cossey University of Akron Conference on Character Theory of Finite Groups in honor of I.M. Isaacs

Normal subgroups and a character correspondence in groups of odd order

Alperin weights Groups of odd order

Normal subgroups

Where do normal subgroups fit in?

We now ask the question, how does Navarro’s map behave withrespect to normal subgroups of the odd order group G ?

Suppose N C G and that Q is a p-subgroup of G , and letP = Q ∩ N. Certainly P C Q and NG (Q) ⊆ NG (P).

Since the map ϕ→ ϕ̃ is a bijection, then there is a uniquecharacter ϕ1 ∈ IBrp(NG (P)|Q) such that ϕ̃1 = ϕ̃. In other words,the map ϕ→ ϕ1 is a bijection from IBrp(G |Q) to IBrp(NG (P)|Q).

J.P. Cossey University of Akron Conference on Character Theory of Finite Groups in honor of I.M. Isaacs

Normal subgroups and a character correspondence in groups of odd order

Alperin weights Groups of odd order

Normal subgroups

Where do normal subgroups fit in?

We now ask the question, how does Navarro’s map behave withrespect to normal subgroups of the odd order group G ?

Suppose N C G and that Q is a p-subgroup of G , and letP = Q ∩ N. Certainly P C Q and NG (Q) ⊆ NG (P).

Since the map ϕ→ ϕ̃ is a bijection, then there is a uniquecharacter ϕ1 ∈ IBrp(NG (P)|Q) such that ϕ̃1 = ϕ̃. In other words,the map ϕ→ ϕ1 is a bijection from IBrp(G |Q) to IBrp(NG (P)|Q).

J.P. Cossey University of Akron Conference on Character Theory of Finite Groups in honor of I.M. Isaacs

Normal subgroups and a character correspondence in groups of odd order

Alperin weights Groups of odd order

Normal subgroups

A picture

Gϕ

NG (Q)ϕ̃

NG (P)

NN(P)

bb

bb

N

ϕ1 ∈ IBrp(NG (P)|Q)

Note that ϕ1 is defined by

the previously discussed bijection,

applied to NG (P).

J.P. Cossey University of Akron Conference on Character Theory of Finite Groups in honor of I.M. Isaacs

Normal subgroups and a character correspondence in groups of odd order

Alperin weights Groups of odd order

Normal subgroups

A picture

Gϕ

NG (Q)ϕ̃

NG (P)

NN(P)

bb

bb

N

ϕ1 ∈ IBrp(NG (P)|Q)

Note that ϕ1 is defined by

the previously discussed bijection,

applied to NG (P).

J.P. Cossey University of Akron Conference on Character Theory of Finite Groups in honor of I.M. Isaacs

Normal subgroups and a character correspondence in groups of odd order

Alperin weights Groups of odd order

Normal subgroups

A picture

Gϕ

NG (Q)ϕ̃

NG (P)

NN(P)

bb

bb

N

ϕ1 ∈ IBrp(NG (P)|Q)

Note that ϕ1 is defined by

the previously discussed bijection,

applied to NG (P).

J.P. Cossey University of Akron Conference on Character Theory of Finite Groups in honor of I.M. Isaacs

Normal subgroups and a character correspondence in groups of odd order

Alperin weights Groups of odd order

Normal subgroups

A picture

Gϕ

NG (Q)ϕ̃

NG (P)

NN(P)

bb

bb

N

ϕ1 ∈ IBrp(NG (P)|Q)

Note that ϕ1 is defined by

the previously discussed bijection,

applied to NG (P).

J.P. Cossey University of Akron Conference on Character Theory of Finite Groups in honor of I.M. Isaacs

Normal subgroups and a character correspondence in groups of odd order

Alperin weights Groups of odd order

Normal subgroups

A picture

G

N

ϕ

θ

NG (Q)ϕ̃

NG (P)

NN(P)

bb

bb

ϕ1

θ̃

J.P. Cossey University of Akron Conference on Character Theory of Finite Groups in honor of I.M. Isaacs

Normal subgroups and a character correspondence in groups of odd order

Alperin weights Groups of odd order

Normal subgroups

A picture

G

N

ϕ

θ

NG (Q)ϕ̃

NG (P)

NN(P)

bb

bb

ϕ1

θ̃

J.P. Cossey University of Akron Conference on Character Theory of Finite Groups in honor of I.M. Isaacs

Normal subgroups and a character correspondence in groups of odd order

Alperin weights Groups of odd order

Normal subgroups

The main result

Using results of Navarro, we can prove the following:

Theorem

(C, 2009) Let G be a group of odd order, N C G , and Q ap-subgroup of G with P = Q ∩ N. If ϕ ∈ IBrp(G |Q) andθ ∈ IBrp(N|P), then

ϕ lies above θ if and only if ϕ1 lies above θ̃.

[ϕN , θ] = [(ϕ1)NN(P), θ̃]

J.P. Cossey University of Akron Conference on Character Theory of Finite Groups in honor of I.M. Isaacs

Normal subgroups and a character correspondence in groups of odd order

Alperin weights Groups of odd order

Normal subgroups

The main result

Using results of Navarro, we can prove the following:

Theorem

(C, 2009) Let G be a group of odd order, N C G , and Q ap-subgroup of G with P = Q ∩ N. If ϕ ∈ IBrp(G |Q) andθ ∈ IBrp(N|P), then

ϕ lies above θ if and only if ϕ1 lies above θ̃.

[ϕN , θ] = [(ϕ1)NN(P), θ̃]

J.P. Cossey University of Akron Conference on Character Theory of Finite Groups in honor of I.M. Isaacs

Normal subgroups and a character correspondence in groups of odd order

Alperin weights Groups of odd order

Normal subgroups

The main result

Using results of Navarro, we can prove the following:

Theorem

(C, 2009) Let G be a group of odd order, N C G , and Q ap-subgroup of G with P = Q ∩ N. If ϕ ∈ IBrp(G |Q) andθ ∈ IBrp(N|P), then

ϕ lies above θ if and only if ϕ1 lies above θ̃.

[ϕN , θ] = [(ϕ1)NN(P), θ̃]

J.P. Cossey University of Akron Conference on Character Theory of Finite Groups in honor of I.M. Isaacs

Normal subgroups and a character correspondence in groups of odd order

Alperin weights Groups of odd order

The proof and applications

Outline of the proof

The proof proceeds in three steps:

(1) Reduce to the case that θ ∈ IBrp(N|P) is invariant in G (whichis possible because the inertia subgroups behave exactly as you’dexpect).

(2) Reduce to the case that G/N is a chief factor, and thus, sinceG is solvable, G/N is cyclic of prime order.

(3) Prove the first statement in this case (the second then followsimmediately). This third step is the most involved.

J.P. Cossey University of Akron Conference on Character Theory of Finite Groups in honor of I.M. Isaacs

Normal subgroups and a character correspondence in groups of odd order

Alperin weights Groups of odd order

The proof and applications

Outline of the proof

The proof proceeds in three steps:

(1) Reduce to the case that θ ∈ IBrp(N|P) is invariant in G (whichis possible because the inertia subgroups behave exactly as you’dexpect).

(2) Reduce to the case that G/N is a chief factor, and thus, sinceG is solvable, G/N is cyclic of prime order.

(3) Prove the first statement in this case (the second then followsimmediately). This third step is the most involved.

J.P. Cossey University of Akron Conference on Character Theory of Finite Groups in honor of I.M. Isaacs

Normal subgroups and a character correspondence in groups of odd order

Alperin weights Groups of odd order

The proof and applications

Outline of the proof

The proof proceeds in three steps:

(1) Reduce to the case that θ ∈ IBrp(N|P) is invariant in G (whichis possible because the inertia subgroups behave exactly as you’dexpect).

(2) Reduce to the case that G/N is a chief factor, and thus, sinceG is solvable, G/N is cyclic of prime order.

(3) Prove the first statement in this case (the second then followsimmediately). This third step is the most involved.

J.P. Cossey University of Akron Conference on Character Theory of Finite Groups in honor of I.M. Isaacs

Normal subgroups and a character correspondence in groups of odd order

Alperin weights Groups of odd order

The proof and applications

Outline of the proof

The proof proceeds in three steps:

(1) Reduce to the case that θ ∈ IBrp(N|P) is invariant in G (whichis possible because the inertia subgroups behave exactly as you’dexpect).

(2) Reduce to the case that G/N is a chief factor, and thus, sinceG is solvable, G/N is cyclic of prime order.

(3) Prove the first statement in this case (the second then followsimmediately). This third step is the most involved.

J.P. Cossey University of Akron Conference on Character Theory of Finite Groups in honor of I.M. Isaacs

Normal subgroups and a character correspondence in groups of odd order

Alperin weights Groups of odd order

The proof and applications

What is this good for?

There are (at least) two reasons one might care about this result:

(1) This shows that the explicit bijection for the Alperin weightconjecture for groups of odd order “preserves” normal subgroups.

(2) This result is used to prove results about the behavior of liftsof Brauer characters in groups of odd order.

J.P. Cossey University of Akron Conference on Character Theory of Finite Groups in honor of I.M. Isaacs

Normal subgroups and a character correspondence in groups of odd order

Alperin weights Groups of odd order

The proof and applications

What is this good for?

There are (at least) two reasons one might care about this result:

(1) This shows that the explicit bijection for the Alperin weightconjecture for groups of odd order “preserves” normal subgroups.

(2) This result is used to prove results about the behavior of liftsof Brauer characters in groups of odd order.

J.P. Cossey University of Akron Conference on Character Theory of Finite Groups in honor of I.M. Isaacs

Normal subgroups and a character correspondence in groups of odd order

Alperin weights Groups of odd order

The proof and applications

What is this good for?

There are (at least) two reasons one might care about this result:

(1) This shows that the explicit bijection for the Alperin weightconjecture for groups of odd order “preserves” normal subgroups.

(2) This result is used to prove results about the behavior of liftsof Brauer characters in groups of odd order.

J.P. Cossey University of Akron Conference on Character Theory of Finite Groups in honor of I.M. Isaacs

Normal subgroups and a character correspondence in groups of odd order