The complex of p-centric and p-radical subgroups and its reduced - - PowerPoint PPT Presentation

The complex of p-centric and p-radical subgroups and its reduced Lefschetz module

John Maginnis and Silvia Onofrei*

Kansas State University The Ohio State University AMS Fall Central Sectional Meeting, University of Akron, Ohio, 20-21 October 2012

Silvia Onofrei (OSU), Properties of Lefschetz modules

Subgroup Complexes in a Finite Group G Subgroup complex

Δ = Δ(C) ∙ 0-simplices:

C = {Q : Q ≤ G} is a collection of subgroups of the group G,

closed under G-conjugation and partially ordered by inclusion

∙ n-simplices: σ = (Q0 < Q1 < ... < Qn), Qi ∈ C

The group G acts by conjugation on the subgroup complex Δ:

∙

isotropy group of σ: Gσ = ∩n

i=0NG(Qi)

∙

fixed point set of Q:

ΔQ = Δ(C Q) with C Q = {P ∈ C∣Q ≤ NG(P)}

Fall Central Sectional Meeting, University of Akron, 20-21 October 2012 1/10

Silvia Onofrei (OSU), Properties of Lefschetz modules

The Reduced Lefschetz Module of a Subgroup Complex in a Finite Group G The reduced Lefschetz virtual module with coefficients in a field k of characteristic p

∙ alternating sum of chain groups: ˜

LG(Δ;k) :=

∣Δ∣

∑

i=−1

(−1)iCi(Δ;k) ∙ element of Green ring of kG: ˜

LG(Δ;k) = ∑

σ∈Δ/G

(−1)∣σ∣IndG

Gσk − k

Fall Central Sectional Meeting, University of Akron, 20-21 October 2012 2/10

Silvia Onofrei (OSU), Properties of Lefschetz modules

The Reduced Lefschetz Module of a Subgroup Complex in a Finite Group G The reduced Lefschetz virtual module with coefficients in a field k of characteristic p

∙ alternating sum of chain groups: ˜

LG(Δ;k) :=

∣Δ∣

∑

i=−1

(−1)iCi(Δ;k) ∙ element of Green ring of kG: ˜

LG(Δ;k) = ∑

σ∈Δ/G

(−1)∣σ∣IndG

Gσk − k Theorem (Robinson, 1988)

Let G be a finite group, k a field of characteristic p and Δ a subgroup complex in G. The number of indecomposable summands of ˜ LG(Δ;k) with vertex Q equals the number of indecomposable summands of ˜ LNG(Q)(ΔQ;k) with vertex Q.

Fall Central Sectional Meeting, University of Akron, 20-21 October 2012 2/10

Silvia Onofrei (OSU), Properties of Lefschetz modules

The Complex of p-Centric and p-Radical Subgroups A nontrivial p-subgroup Q of G is p-radical if Q = Op(NG(Q)) is p-centric if Z(Q) ∈ Sylp(CG(Q))

Fall Central Sectional Meeting, University of Akron, 20-21 October 2012 3/10

Silvia Onofrei (OSU), Properties of Lefschetz modules

The Complex of p-Centric and p-Radical Subgroups A nontrivial p-subgroup Q of G is p-radical if Q = Op(NG(Q)) is p-centric if Z(Q) ∈ Sylp(CG(Q))

Dp(G) complex

∙ collection of p-centric p-radical subgroups of G

Dwyer(1997)

∙ best candidate for a p-local geometry

Smith, Yoshiara(1997)

∙ used in cohomology decompositions

Dwyer(1998), Grodal(2001) Benson, Smith(2008)

˜

LG(Dp(G);k)

∙ not indecomposable, not projective ∙ vertices are subgroups of non-centric p-radicals

Sawabe(2005)

Fall Central Sectional Meeting, University of Akron, 20-21 October 2012 3/10

Silvia Onofrei (OSU), Properties of Lefschetz modules

The Complex of p-Centric and p-Radical Subgroups A nontrivial p-subgroup Q of G is p-radical if Q = Op(NG(Q)) is p-centric if Z(Q) ∈ Sylp(CG(Q))

Dp(G) complex

∙ collection of p-centric p-radical subgroups of G

Dwyer(1997)

∙ best candidate for a p-local geometry

Smith, Yoshiara(1997)

∙ used in cohomology decompositions

Dwyer(1998), Grodal(2001) Benson, Smith(2008)

˜

LG(Dp(G);k)

∙ not indecomposable, not projective ∙ vertices are subgroups of non-centric p-radicals

Sawabe(2005)

If G is a finite simple group of Lie type then ˜ LG(Dp(G);k) ≃ StG the irreducible and projective Steinberg module.

Fall Central Sectional Meeting, University of Akron, 20-21 October 2012 3/10

Silvia Onofrei (OSU), Properties of Lefschetz modules

Terminology and Notation: Groups G is a finite group and p a prime divisor of its order a p-local subgroup is the normalizer of a finite p-subgroup of G a p-central element is an element in the center of a Sylow p-subgroup of G kG is the group algebra with k a field of characteristic p

Fall Central Sectional Meeting, University of Akron, 20-21 October 2012 4/10

Silvia Onofrei (OSU), Properties of Lefschetz modules

Terminology and Notation: Groups G is a finite group and p a prime divisor of its order a p-local subgroup is the normalizer of a finite p-subgroup of G a p-central element is an element in the center of a Sylow p-subgroup of G kG is the group algebra with k a field of characteristic p

1

G has characteristic p if CG(Op(G)) ≤ Op(G)

2

G has local characteristic p if all p-local subgroups of G have characteristic p

3

G has parabolic characteristic p if all p-local subgroups which contain a Sylow p-subgroup

• f G have characteristic p

Fall Central Sectional Meeting, University of Akron, 20-21 October 2012 4/10

Silvia Onofrei (OSU), Properties of Lefschetz modules

No Vertex of ˜ LG(Dp(G);k) Contains p-Central Elements

Proposition (Maginnis, Onofrei, 2009)

Assume G is a finite group of parabolic characteristic p. Suppose that t is an element of order p in G such that Op(CG(t)) contains p-central elements. Then no vertex of the reduced Lefschetz module ˜ LG(Dp(G);k) contains a conjugate of t.

Fall Central Sectional Meeting, University of Akron, 20-21 October 2012 5/10

Silvia Onofrei (OSU), Properties of Lefschetz modules

No Vertex of ˜ LG(Dp(G);k) Contains p-Central Elements

Proposition (Maginnis, Onofrei, 2009)

Assume G is a finite group of parabolic characteristic p. Suppose that t is an element of order p in G such that Op(CG(t)) contains p-central elements. Then no vertex of the reduced Lefschetz module ˜ LG(Dp(G);k) contains a conjugate of t. Sketch of proof: Set T := ⟨t⟩.

Fall Central Sectional Meeting, University of Akron, 20-21 October 2012 5/10

Silvia Onofrei (OSU), Properties of Lefschetz modules

No Vertex of ˜ LG(Dp(G);k) Contains p-Central Elements

Proposition (Maginnis, Onofrei, 2009)

Assume G is a finite group of parabolic characteristic p. Suppose that t is an element of order p in G such that Op(CG(t)) contains p-central elements. Then no vertex of the reduced Lefschetz module ˜ LG(Dp(G);k) contains a conjugate of t. Sketch of proof: Set T := ⟨t⟩. If Op(CG(T)) contains p-central elements then Dp(G)T is NG(T)-contractible.

Fall Central Sectional Meeting, University of Akron, 20-21 October 2012 5/10

Silvia Onofrei (OSU), Properties of Lefschetz modules

No Vertex of ˜ LG(Dp(G);k) Contains p-Central Elements

Proposition (Maginnis, Onofrei, 2009)

Assume G is a finite group of parabolic characteristic p. Suppose that t is an element of order p in G such that Op(CG(t)) contains p-central elements. Then no vertex of the reduced Lefschetz module ˜ LG(Dp(G);k) contains a conjugate of t. Sketch of proof: Set T := ⟨t⟩. If Op(CG(T)) contains p-central elements then Dp(G)T is NG(T)-contractible. Thus D(G)T is mod-p acyclic.

Fall Central Sectional Meeting, University of Akron, 20-21 October 2012 5/10

Silvia Onofrei (OSU), Properties of Lefschetz modules

No Vertex of ˜ LG(Dp(G);k) Contains p-Central Elements

Proposition (Maginnis, Onofrei, 2009)

Assume G is a finite group of parabolic characteristic p. Suppose that t is an element of order p in G such that Op(CG(t)) contains p-central elements. Then no vertex of the reduced Lefschetz module ˜ LG(Dp(G);k) contains a conjugate of t. Sketch of proof: Set T := ⟨t⟩. If Op(CG(T)) contains p-central elements then Dp(G)T is NG(T)-contractible. Thus D(G)T is mod-p acyclic. And P .A. Smith theory: D(G)Q is mod-p acyclic for any p-subgroup Q > T.

Fall Central Sectional Meeting, University of Akron, 20-21 October 2012 5/10

Silvia Onofrei (OSU), Properties of Lefschetz modules

No Vertex of ˜ LG(Dp(G);k) Contains p-Central Elements

Proposition (Maginnis, Onofrei, 2009)

Assume G is a finite group of parabolic characteristic p. Suppose that t is an element of order p in G such that Op(CG(t)) contains p-central elements. Then no vertex of the reduced Lefschetz module ˜ LG(Dp(G);k) contains a conjugate of t. Sketch of proof: Set T := ⟨t⟩. If Op(CG(T)) contains p-central elements then Dp(G)T is NG(T)-contractible. Thus D(G)T is mod-p acyclic. And P .A. Smith theory: D(G)Q is mod-p acyclic for any p-subgroup Q > T. It follows ˜ LNG(Q)(D(G)Q;k) = 0.

Fall Central Sectional Meeting, University of Akron, 20-21 October 2012 5/10

Silvia Onofrei (OSU), Properties of Lefschetz modules

No Vertex of ˜ LG(Dp(G);k) Contains p-Central Elements

Proposition (Maginnis, Onofrei, 2009)

Assume G is a finite group of parabolic characteristic p. Suppose that t is an element of order p in G such that Op(CG(t)) contains p-central elements. Then no vertex of the reduced Lefschetz module ˜ LG(Dp(G);k) contains a conjugate of t. Sketch of proof: Set T := ⟨t⟩. If Op(CG(T)) contains p-central elements then Dp(G)T is NG(T)-contractible. Thus D(G)T is mod-p acyclic. And P .A. Smith theory: D(G)Q is mod-p acyclic for any p-subgroup Q > T. It follows ˜ LNG(Q)(D(G)Q;k) = 0. An application of Robinson’s theorem gives the result.

Fall Central Sectional Meeting, University of Akron, 20-21 October 2012 5/10

Silvia Onofrei (OSU), Properties of Lefschetz modules

No Vertex of ˜ LG(Dp(G);k) Contains p-Central Elements

Proposition (Maginnis, Onofrei, 2009)

Assume G is a finite group of parabolic characteristic p. Suppose that t is an element of order p in G such that Op(CG(t)) contains p-central elements. Then no vertex of the reduced Lefschetz module ˜ LG(Dp(G);k) contains a conjugate of t. Sketch of proof: Set T := ⟨t⟩. If Op(CG(T)) contains p-central elements then Dp(G)T is NG(T)-contractible. Thus D(G)T is mod-p acyclic. And P .A. Smith theory: D(G)Q is mod-p acyclic for any p-subgroup Q > T. It follows ˜ LNG(Q)(D(G)Q;k) = 0. An application of Robinson’s theorem gives the result.

Fall Central Sectional Meeting, University of Akron, 20-21 October 2012 5/10

Silvia Onofrei (OSU), Properties of Lefschetz modules

No Vertex of ˜ LG(Dp(G);k) Contains p-Central Elements

Proposition (Maginnis, Onofrei, 2009)

Assume G is a finite group of parabolic characteristic p. Suppose that t is an element of order p in G such that Op(CG(t)) contains p-central elements. Then no vertex of the reduced Lefschetz module ˜ LG(Dp(G);k) contains a conjugate of t. Sketch of proof: Set T := ⟨t⟩. If Op(CG(T)) contains p-central elements then Dp(G)T is NG(T)-contractible. Thus D(G)T is mod-p acyclic. And P .A. Smith theory: D(G)Q is mod-p acyclic for any p-subgroup Q > T. It follows ˜ LNG(Q)(D(G)Q;k) = 0. An application of Robinson’s theorem gives the result.

Fall Central Sectional Meeting, University of Akron, 20-21 October 2012 5/10

Silvia Onofrei (OSU), Properties of Lefschetz modules

Fixed Point Sets of Purely Noncentral p-Subgroups

Theorem (Maginnis, Onofrei, 2012)

Let G be a finite group of parabolic characteristic p and let T be a p-subgroup of G. Assume the following hold: (N1) The group OC := Op(TCG(T)) is purely noncentral in G. (N2) C := TCG(T) = OC.H.K where H has parabolic characteristic p and L := OC.H is normal in NG(T). (N3) A Sylow p-subgroup of L contains p-central elements of G. Then there is an NG(T)-equivariant homotopy equivalence between Dp(G)T and Dp(H).

Fall Central Sectional Meeting, University of Akron, 20-21 October 2012 6/10

Silvia Onofrei (OSU), Properties of Lefschetz modules

Main Theorem on Vertices of ˜ LG(Dp(G);k)

Theorem (Maginnis, Onofrei, 2012)

Let G be a finite group of parabolic characteristic p and let T be a p-subgroup of G. Assume that the following conditions hold:

(i). C := TCG(T) = OC.H.K where OC = Op(C)

and L := OC.H is the generalized Fitting subgroup of C.

(ii). The group H = L/OC is a finite simple group of Lie type in characteristic p. (iii). A Sylow p-subgroup of L contains p-central elements of G.

Then T is a vertex of an indecomposable summand of ˜ LG(Dp(G);k) if and only if:

(a). T = OC is purely noncentral. (b). K = C/L is a p′-group. (c). ∣NG(T)/C∣ is relatively prime to p.

Under these conditions, there will exist a unique indecomposable summand of ˜ LG(Dp(G);k) with vertex T, which will lie in a block of kG with defect group T.

Fall Central Sectional Meeting, University of Akron, 20-21 October 2012 7/10

Silvia Onofrei (OSU), Properties of Lefschetz modules

Main Theorem: Sketch of the Argument T is a vertex of ˜ LG(Dp(G);k) if and only if T is a vertex of ˜ LNG(T)(Dp(G)T ;k)

Fall Central Sectional Meeting, University of Akron, 20-21 October 2012 8/10

Silvia Onofrei (OSU), Properties of Lefschetz modules

Main Theorem: Sketch of the Argument T is a vertex of ˜ LG(Dp(G);k) if and only if T is a vertex of ˜ LNG(T)(Dp(G)T ;k) OC = Op(C) = OC(TCG(T)) is noncentral

Fall Central Sectional Meeting, University of Akron, 20-21 October 2012 8/10

Silvia Onofrei (OSU), Properties of Lefschetz modules

Main Theorem: Sketch of the Argument T is a vertex of ˜ LG(Dp(G);k) if and only if T is a vertex of ˜ LNG(T)(Dp(G)T ;k) OC = Op(C) = OC(TCG(T)) is noncentral NG(T) = OC.H.K ′ with K ′ = NG(T)/L and L = OC.H

Fall Central Sectional Meeting, University of Akron, 20-21 October 2012 8/10

Silvia Onofrei (OSU), Properties of Lefschetz modules

Main Theorem: Sketch of the Argument T is a vertex of ˜ LG(Dp(G);k) if and only if T is a vertex of ˜ LNG(T)(Dp(G)T ;k) OC = Op(C) = OC(TCG(T)) is noncentral NG(T) = OC.H.K ′ with K ′ = NG(T)/L and L = OC.H

Dp(G)T is NG(T)-equivariantly homotopy equivalent to Dp(H)

Fall Central Sectional Meeting, University of Akron, 20-21 October 2012 8/10

Silvia Onofrei (OSU), Properties of Lefschetz modules

Main Theorem: Sketch of the Argument T is a vertex of ˜ LG(Dp(G);k) if and only if T is a vertex of ˜ LNG(T)(Dp(G)T ;k) OC = Op(C) = OC(TCG(T)) is noncentral NG(T) = OC.H.K ′ with K ′ = NG(T)/L and L = OC.H

Dp(G)T is NG(T)-equivariantly homotopy equivalent to Dp(H) Dp(H) is NG(T)-equivariantly homotopy equivalent to the Tits building Δ of H

Fall Central Sectional Meeting, University of Akron, 20-21 October 2012 8/10

Silvia Onofrei (OSU), Properties of Lefschetz modules

Main Theorem: Sketch of the Argument T is a vertex of ˜ LG(Dp(G);k) if and only if T is a vertex of ˜ LNG(T)(Dp(G)T ;k) OC = Op(C) = OC(TCG(T)) is noncentral NG(T) = OC.H.K ′ with K ′ = NG(T)/L and L = OC.H

Dp(G)T is NG(T)-equivariantly homotopy equivalent to Dp(H) Dp(H) is NG(T)-equivariantly homotopy equivalent to the Tits building Δ of H

˜

LNG(T)(Dp(G)T ;k) = M is an irreducible kNG(T)-module, it is the inflation to NG(T) of the extended Steinberg module for H.K ′

Fall Central Sectional Meeting, University of Akron, 20-21 October 2012 8/10

Silvia Onofrei (OSU), Properties of Lefschetz modules

Main Theorem: Sketch of the Argument T is a vertex of ˜ LG(Dp(G);k) if and only if T is a vertex of ˜ LNG(T)(Dp(G)T ;k) OC = Op(C) = OC(TCG(T)) is noncentral NG(T) = OC.H.K ′ with K ′ = NG(T)/L and L = OC.H

Dp(G)T is NG(T)-equivariantly homotopy equivalent to Dp(H) Dp(H) is NG(T)-equivariantly homotopy equivalent to the Tits building Δ of H

˜

LNG(T)(Dp(G)T ;k) = M is an irreducible kNG(T)-module, it is the inflation to NG(T) of the extended Steinberg module for H.K ′ using that kH has two blocks, the principal block and a block of defect zero, and that L = F∗(C), the generalized Fitting subgroup of C, we deduce that M lies in a nonprincipal block of kNG(T) with defect group OC.S′ ≃ vx(M) and with S′ a Sylow p-subgroup of K ′

Fall Central Sectional Meeting, University of Akron, 20-21 October 2012 8/10

Silvia Onofrei (OSU), Properties of Lefschetz modules

Main Theorem: Sketch of the Argument T is a vertex of ˜ LG(Dp(G);k) if and only if T is a vertex of ˜ LNG(T)(Dp(G)T ;k) OC = Op(C) = OC(TCG(T)) is noncentral NG(T) = OC.H.K ′ with K ′ = NG(T)/L and L = OC.H

Dp(G)T is NG(T)-equivariantly homotopy equivalent to Dp(H) Dp(H) is NG(T)-equivariantly homotopy equivalent to the Tits building Δ of H

˜

LNG(T)(Dp(G)T ;k) = M is an irreducible kNG(T)-module, it is the inflation to NG(T) of the extended Steinberg module for H.K ′ using that kH has two blocks, the principal block and a block of defect zero, and that L = F∗(C), the generalized Fitting subgroup of C, we deduce that M lies in a nonprincipal block of kNG(T) with defect group OC.S′ ≃ vx(M) and with S′ a Sylow p-subgroup of K ′ since T ≤ OC, we have T ≃ vx(M) if and only if T = OC and S′ = 1

Fall Central Sectional Meeting, University of Akron, 20-21 October 2012 8/10

Silvia Onofrei (OSU), Properties of Lefschetz modules

Examples Involving Sporadic Simple Groups in Characteristic 3

Fall Central Sectional Meeting, University of Akron, 20-21 October 2012 9/10

Silvia Onofrei (OSU), Properties of Lefschetz modules

Examples Involving Sporadic Simple Groups in Characteristic 3

G CG(t) = O3(CG(t)).Ht.Kt Ht

D3(G)t

T TCG(T) NG(T)

D3(G)T

Fi′

24

C(3A) = 3× O+

8 (3) : 3

O+

8 (3)

D4 C(3C) = 37.2.U4(3) point C(3D) = 32+4+6.(A4 × 2A4) point C(3E) = 32 × G2(3) G2(3) G2 32 32 × G2(3)

(32 : 2× G2(3)).2

G2

Fall Central Sectional Meeting, University of Akron, 20-21 October 2012 9/10

Silvia Onofrei (OSU), Properties of Lefschetz modules

Examples Involving Sporadic Simple Groups in Characteristic 3

G CG(t) = O3(CG(t)).Ht.Kt Ht

D3(G)t

T TCG(T) NG(T)

D3(G)T

Fi′

24

C(3A) = 3× O+

8 (3) : 3

O+

8 (3)

D4 32 32 × G2(3)

(32 : 2× G2(3)).2

G2 C(3C) = 37.2.U4(3) point C(3D) = 32+4+6.(A4 × 2A4) point C(3E) = 32 × G2(3) G2(3) G2 32 32 × G2(3)

(32 : 2× G2(3)).2

G2

Fall Central Sectional Meeting, University of Akron, 20-21 October 2012 9/10

Silvia Onofrei (OSU), Properties of Lefschetz modules

Examples Involving Sporadic Simple Groups in Characteristic 3

G CG(t) = O3(CG(t)).Ht.Kt Ht

D3(G)t

T TCG(T) NG(T)

D3(G)T

Fi′

24

C(3A) = 3× O+

8 (3) : 3

O+

8 (3)

D4 32 32 × G2(3)

(32 : 2× G2(3)).2

G2 C(3C) = 37.2.U4(3) point C(3D) = 32+4+6.(A4 × 2A4) point C(3E) = 32 × G2(3) G2(3) G2 32 32 × G2(3)

(32 : 2× G2(3)).2

G2 Th C(3A) = 3× G2(3) G2(3) G2 3 3× G2(3)

(3× G2(3)) : 2

G2 C(3C) = 3× 34 : 2A6 point

Fall Central Sectional Meeting, University of Akron, 20-21 October 2012 9/10

Silvia Onofrei (OSU), Properties of Lefschetz modules

Examples Involving Sporadic Simple Groups in Characteristic 3

G CG(t) = O3(CG(t)).Ht.Kt Ht

D3(G)t

T TCG(T) NG(T)

D3(G)T

Fi′

24

C(3A) = 3× O+

8 (3) : 3

O+

8 (3)

D4 32 32 × G2(3)

(32 : 2× G2(3)).2

G2 C(3C) = 37.2.U4(3) point C(3D) = 32+4+6.(A4 × 2A4) point C(3E) = 32 × G2(3) G2(3) G2 32 32 × G2(3)

(32 : 2× G2(3)).2

G2 Th C(3A) = 3× G2(3) G2(3) G2 3 3× G2(3)

(3× G2(3)) : 2

G2 C(3C) = 3× 34 : 2A6 point G CG(t) = O3(CG(t)).Ht.Kt Ht

D3(G)t

T TCG(T) NG(T)

D3(G)T

M C(3A) = 3.Fi′

24

Fi′

24

D3(Fi′

24)

31+2 31+2 × G2(3)

(31+2 : 22 × G2(3)).2

G2 C(3C) = 3× Th Th

D3(Th)

32 32 × G2(3)

(31+2 : 2× G2(3)).2

G2

Fall Central Sectional Meeting, University of Akron, 20-21 October 2012 9/10

Silvia Onofrei (OSU), Properties of Lefschetz modules

Thank You

Fall Central Sectional Meeting, University of Akron, 20-21 October 2012 10/10

Silvia Onofrei (OSU), Properties of Lefschetz modules

Thank You The End

Fall Central Sectional Meeting, University of Akron, 20-21 October 2012 10/10