Collections of distinguished p-subgroups and homology - - PDF document

Collections of distinguished p-subgroups and homology decompositions for classifying spaces of finite groups Silvia Onofrei

Department of Mathematics University of California at Riverside

• geometries for the sporadic simple groups
• Buekenhout (mid 70’s)
• Ronan and Smith (1980)
• Ronan and Stroth (1984)
• collections of subgroups related to group cohomology
• Brown (1975)
• Quillen (1978)
• Bouc (1984)
• homotopy equivalences for each sporadic group, be-

tween a 2-local geometry and the simplicial complex of a collection of 2-subgroups

• Benson and Smith (2004)

An example: the sporadic group Co3

(∆)

• 2

P

• 22

L

• 24

M

GP = 2.S6(2) GL = 22+63.(S3 × S3) GM = 24L4(2)

• ∆ ∼ B2(Co3)

and ∆ ∼ Bcen

2 (Co3)

• define: distinguished 2-radical subgroups

contain involutions of central type in their centers

• ∆ ∼

B2(Co3)

Standard collections of p-subgroups

A collection Cp(G) of p-subgroups of G

• set of p-subgroups which is closed under conjugation
• a G-poset under the inclusion relation

|Cp(G)| - the corresponding simplicial complex

• Quillen collection

Ap(G) = {E | 1 = E elementary abelian p-subgroup}

• Brown collection

Sp(G) = {P | P nontrivial p-subgroup of G}

• Bouc collection

Bp(G) = {R | 1 = R, R = OpNG(R)}

• The collection of p-centric and p-radical subgroups

Bcen

p (G) = {P | P ∈ Bp(G), Z(P) ∈ Sylp(CG(P))}

Poset homotopy

• For X a G-poset.
• X>x = {y ∈ X | y > x}
• |X| the associated simplicial complex
• X H the fixed point set of H ≤ G

A poset X is conically contractible if there is a poset map f : X → X and an element x0 ∈ X such that x ≤ f(x) ≥ x0, ∀x ∈ X

• r

x ≥ f(x) ≤ x0, ∀x ∈ X Theorem 1 [Brdn]. Suppose that X and Y are two finite G-posets and f : |X| → |Y| is a G-map. The poset map f is a G-homotopy equivalence if and only if, for all subgroups H ≤ G, the map f re- stricts to an ordinary homotopy equivalence f H : |X|H → |Y|H. Theorem 2 [ThWb] Let G be a group, let X, Y be G-posets and let f : X → Y be a map of G-posets. Suppose that either: (i) for all y ∈ Y, f −1 Y≤y

• is Gy-contractible or

(ii) for all y ∈ Y, f −1 Y≥y

• is Gy-contractible.

Then f is a G-homotopy equivalence.

• Proposition. [ThWb] Let Y be a G-poset and X a G-invariant sub-

poset of Y such that for each y ∈ Y \ X, the subposet Y<y (or dually Y>y) is Gy-contractible. Then the inclusion X → Y is a G-homotopy equivalence.

Homology decompositions

• EG the universal cover of BG
• BG = EG/G the classifying space of G
• X a G-space
• X ×G EG = (X × EG)/G the Borel construction of X
• Homology decomposition

Y − → BG Usually Y = X ×G EG

• A collection is ample if and only if the natural map

| C | ×GEG → BG induces an isomorphism in mod p homology.

• Sharpness
• acyclicity for Bredon homology
• categories: EAC, EOC, C.

• C
• objects: the subgroups H ∈ C
• morphisms: the inclusion maps
• G-action: conjugation
• isotropy groups: NG(H), H ∈ C
• EAC
• objects: (H, i), i : H → G monomorphism, i(H) ∈ C
• morphisms: arrows (H, i) → (K, j),

group homomorphism ρ : H → K and jρ = i

• G-action: g · (H, i) = (H, cgi), cg : G → G

with cg(x) = gxg−1.

• isotropy groups: CG(i(H)), i(H) ∈ C
• EOC
• objects: (G/H, xH), H ∈ C, xH ∈ G/H
• morphisms: G-maps f : G/H → G/K

such that f(xH) = yK

• G-action:

g · (G/H, xH) = (G/H, gxH)

• isotropy groups: xH = xHx−1 with H ∈ C

p-Local Structure

The group G has characteristic p if CG(Op(G)) ≤ Op(G) If all p-local subgroups of G have characteristic p then G has local characteristic p.

• a. Let G be a group of characteristic p. Then G has

local characteristic p.

• b. Assume that G has characteristic p and that H is a

normal subgroup of G. Then H has characteristic p.

• c. Let G be of local characteristic p and P a non-trivial

p-subgroup of G. Then CG(P) is of characteristic p. A parabolic subgroup is a subgroup of G which contains a Sylow p-subgroup of G. G has parabolic characteristic p if all p-local, parabolic subgroups of G have characteristic p.

p-Distinguished Subgroups

Γp(G) = the elements of order p of central type in G For a p-subgroup P of G define:

• P = x|x ∈ Ω1Z(P) ∩ Γp(G)

For Cp(G) a collection of p-subgroups of G denote:

• Cp(G) = {P|P ∈ Cp(G) and

P = 1 }

• For any G:

Bcen

p (G) ⊆

Bp(G).

• If G has local characteristic p:

Bp(G) = Bp(G) = Bcen

p (G).

Proposition: Let G be a group of parabolic characteristic p and let R ∈ Bp(G). Then: a). CG(R) = Z(R); b). NG(R) has characteristic p; c). R is a p-centric subgroup of G. Consequently Bp(G) = Bcen

p (G).

Let R ∈ Bp(G) so R = OpNG(R) and let L = CG(R). As R is p-radical: Op(L) = Z(R) Let z ∈ Z(R) ∩ Γp(G). Then CG(z) contains a Sylow p-subgroup of G. NG(z) and CG(z) have characteristic p. But R ≤ CG(z) and L ≤ CG(z), thus CCG(z)(R) = CG(z) ∩ L = L has characteristic p. Therefore CL(Op(L)) ≤ Op(L). This gives CL(Z(R)) = L ≤ Z(R) and Z(R) = L. Finally: CNG(R)(OpNG(R)) = L ≤ R

We formulate the following conditions: (M) Given P ∈ Sp(G); the subgroup NG(P) is contained in a parabolic p-local subgroup of G. (Cl) The elements of order p and of central type in G are closed under products of commuting elements. (Ch) The group G has local characteristic p. (PCh) The group G has parabolic characteristic p.

Theorem A

Let C be one of the collections Ap(G), Sp(G) or Bp(G). Then there exist homotopy equivalences, summarized in the following table:

• Ap
• Sp
• Bp

|EOC|

• · · ·· · ·· · ·· · ·· · ·· · ·· · ·· · ·· · ·
• (Cl, M, PCh)

|C|

• .

. . . . . . . . . . . . . . . . .

• |

| | |

• |

| | | (Cl, M, PCh)

|EAC|

• |

| | |

• |

| | | (Cl, M, PCh) (Cl, PCh) (Cl, PCh) (Cl, Ch) (Cl, Ch) · · ·· · ·· · ·· · ·· · ·· · ·· · ·· · ·

• .

. . . . . . . . . . . . . . . . .

Notation:

• A solid line corresponds to a G-homotopy equivalence.
• A dashed line corresponds to a S-homotopy equivalence.
• A dotted line corresponds to an ordinary homotopy equivalence.
• S denotes a Sylow p-subgroup of G.
• A label (c) means that the corresponding homotopy equivalence

holds under hypothesis (c).

Excerpts of Proof

• Assume that G is a finite group with the property that

(M) holds, then Bp(G) ∼G Sp(G). ⊚ If we show that for each P ∈ Sp(G) \ Bp(G), the sub- poset Sp(G)>P is NG(P)-contractible, then the inclusion

• Bp(G) ֒

→ Sp(G) is a G-homotopy equivalence. ⊚ Let P ∈ Sp(G) and Q ∈ Sp(G)>P. Then NQ(P) is a distinguished p-subgroup in G. Note that P < NQ(P) ≤ Q and that Z(Q) ≤ Z(NQ(P)). ⊚ Let P ∈ Sp(G) \ Bp(G) and set ONP = Op(NG(P)). Let Q ∈ Sp(G)>P so P < NQ(P) ≤ Q. For ¯ S ∈ Sylp(NG(P)) with NQ(P) ≤ ¯ S, let S ∈ Sylp(G) such that ¯ S = S ∩ NG(P).

⊚ M - p-local subgroup of G with NG(P) ≤ M |M|p = |G|p and R = Op(M) ⊚ Consider the following string of NG(P)-equivariant poset maps Sp(G)>P → Sp(G)>P Q ≥ NQ(P) ≤ NQ(P)NR(P)ONP ≥ NR(P)ONP The group products are p-subgroups in G: ⋄ R M ≥ NG(P) ≥ NQ(P) ⋄ ONP NG(P) ≥ NQ(P) The above p-subgroups are distinguished: ⋄ 1 = Z(S) ∩ R ≤ Z(R) ≤ Z(NR(P)ONP) ⋄ 1 = Z(S) ∩ Z(R) ≤ Z(NQ(P)NR(P)ONP) It follows that Sp(G)>P is NG(P)-contractible.

Theorem B

The collections Ap(G), Bp(G), Sp(G) are ample.

• Ap(G) is centralizer and normalizer sharp.
• Bp(G) is subgroup and normalizer sharp.
• Sp(G) is subgroup, centralizer and normalizer sharp.

A simple example: the group GL(3, F2)

Webb’s alternating-sum formula for group cohomology: H∗(G; Fp) =

• σ∈∆/G

(−1)dimσH∗(Gσ; Fp) Let G = GL(3, F2). S4

• D8
• S4

The Webb alternating sum formula for the building, writ- ten as an exact sequence for cohomology:

0 → H∗(G, F2) → H∗(S4, F2) ⊕ H∗(S4, F2) → H∗(D8, F2) → 0

The homology approximation for BG: (BD8 × I) ∐ BS4 ∐ BS4/(identifications)

The Quillen complex There is an exact sequence in cohomology:

0 → H∗(G, F2) → H∗(D8, F2) ⊕ H∗(S4, F2) ⊕ H∗(S4, F2) → → H∗(D8, F2) ⊕ H∗(D8, F2) → 0

The normalizer decomposition for A2(G) is sharp.