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 Co 3 2 2 2 4 2 (∆) • • • P L M � G P = 2 . S 6 (2) G L = 2 2+6 3 . ( S 3 × S 3 ) G M = 2 4 L 4 (2) ∆ �∼ B cen • ∆ �∼ B 2 (Co 3 ) and 2 (Co 3 ) • define : distinguished 2 -radical subgroups contain involutions of central type in their centers • ∆ ∼ � B 2 (Co 3 )

Standard collections of p-subgroups A collection C p ( G ) of p -subgroups of G • set of p -subgroups which is closed under conjugation • a G -poset under the inclusion relation |C p ( G ) | - the corresponding simplicial complex • Quillen collection A p ( G ) = { E | 1 � = E elementary abelian p-subgroup } • Brown collection S p ( G ) = { P | P nontrivial p-subgroup of G } • Bouc collection B p ( G ) = { R | 1 � = R, R = O p N G ( R ) } • The collection of p -centric and p -radical subgroups B cen p ( G ) = { P | P ∈ B p ( G ) , Z ( P ) ∈ Syl p ( C G ( 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 x 0 ∈ X such that x ≤ f ( x ) ≥ x 0 , ∀ x ∈ X x ≥ f ( x ) ≤ x 0 , ∀ x ∈ X or 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 G y -contractible or (ii) for all y ∈ Y , f − 1 � � Y ≥ y is G y -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 G y -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 Y = X × G EG Usually • A collection is ample if and only if the natural map | C | × G EG → BG induces an isomorphism in mod p homology. • Sharpness ◦ acyclicity for Bredon homology ◦ categories: E A C , E O C , C .

• C ◦ objects: the subgroups H ∈ C ◦ morphisms: the inclusion maps ◦ G -action: conjugation ◦ isotropy groups: N G ( H ) , H ∈ C • E A C ◦ 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, c g i ) , c g : G → G with c g ( x ) = gxg − 1 . ◦ isotropy groups: C G ( i ( H )) , i ( H ) ∈ C • E O C ◦ 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: x H = xHx − 1 with H ∈ C

p-Local Structure The group G has characteristic p if C G ( O p ( G )) ≤ O p ( 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 C G ( 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 Γ p ( G ) = the elements of order p of central type in G For a p -subgroup P of G define: � P = � x | x ∈ Ω 1 Z ( P ) ∩ Γ p ( G ) � For C p ( G ) a collection of p -subgroups of G denote: C p ( G ) = { P | P ∈ C p ( G ) and � � P � = 1 } p ( G ) ⊆ � B cen • For any G : B p ( G ) . • If G has local characteristic p : B p ( G ) = � B p ( G ) = B cen p ( G ) .

Proposition : Let G be a group of parabolic characteristic p and let R ∈ � B p ( G ) . Then: a). C G ( R ) = Z ( R ) ; b). N G ( R ) has characteristic p ; c). R is a p -centric subgroup of G . Consequently � B p ( G ) = B cen p ( G ) . Let R ∈ � B p ( G ) so R = O p N G ( R ) and let L = C G ( R ) . As R is p -radical: O p ( L ) = Z ( R ) Let z ∈ Z ( R ) ∩ Γ p ( G ) . Then C G ( z ) contains a Sylow p -subgroup of G . N G ( � z � ) and C G ( z ) have characteristic p . But R ≤ C G ( z ) and L ≤ C G ( z ) , thus C C G ( z ) ( R ) = C G ( z ) ∩ L = L has characteristic p . Therefore C L ( O p ( L )) ≤ O p ( L ) . This gives C L ( Z ( R )) = L ≤ Z ( R ) and Z ( R ) = L . Finally: C N G ( R ) ( O p N G ( R )) = L ≤ R

We formulate the following conditions: ( M ) Given P ∈ � S p ( G ) ; the subgroup N G ( P ) is contained in a parabolic p -local subgroup of G . ( C l ) The elements of order p and of central type in G are closed under products of commuting elements. ( C h ) The group G has local characteristic p . ( PC h ) The group G has parabolic characteristic p .

Theorem A Let C be one of the collections � A p ( G ) , � S p ( G ) or � B p ( G ) . Then there exist homotopy equivalences, summarized in the following table: � � � A p S p B p ( C l, M , PC h ) | E O C | •· · ·· · ·· · ·· · ·· · ·· · ·· · ·· · ·· · · • • . . | | . . . . ( C l, PC h ) ( C l, PC h ) . | | . . . . | | . . . . ( C l, M , PC h ) . | | . . |C| . . • • • . . | | . . . . . ( C l, C h ) ( C l, C h ) | | . . . . | | . . . . ( C l, M , PC h ) . | | | E A C | • • · · ·· · ·· · ·· · ·· · ·· · ·· · ·· · · • 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 � B p ( G ) ∼ G � S p ( G ) . ⊚ If we show that for each P ∈ � S p ( G ) \ � B p ( G ) , the sub- poset � S p ( G ) >P is N G ( P ) -contractible, then the inclusion � → � B p ( G ) ֒ S p ( G ) is a G -homotopy equivalence. ⊚ Let P ∈ S p ( G ) and Q ∈ � S p ( G ) >P . Then N Q ( P ) is a distinguished p -subgroup in G . Note that P < N Q ( P ) ≤ Q and that Z ( Q ) ≤ Z ( N Q ( P )) . ⊚ Let P ∈ � S p ( G ) \ � B p ( G ) and set O NP = O p ( N G ( P )) . Let Q ∈ � S p ( G ) >P so P < N Q ( P ) ≤ Q . For ¯ S ∈ Syl p ( N G ( P )) with N Q ( P ) ≤ ¯ S , let S ∈ Syl p ( G ) such that ¯ S = S ∩ N G ( P ) .

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

Theorem B The collections � A p ( G ) , � B p ( G ) , � S p ( G ) are ample. � A p ( G ) is centralizer and normalizer sharp. � B p ( G ) is subgroup and normalizer sharp. � S p ( G ) is subgroup, centralizer and normalizer sharp.

A simple example: the group GL (3 , F 2 ) Webb’s alternating-sum formula for group cohomology: � H ∗ ( G ; F p ) = ( − 1) dim σ H ∗ ( G σ ; F p ) σ ∈ ∆ /G Let G = GL (3 , F 2 ) . S 4 S 4 D 8 � � The Webb alternating sum formula for the building, writ- ten as an exact sequence for cohomology: 0 → H ∗ ( G, F 2 ) → H ∗ ( S 4 , F 2 ) ⊕ H ∗ ( S 4 , F 2 ) → H ∗ ( D 8 , F 2 ) → 0 The homology approximation for BG : ( BD 8 × I ) ∐ BS 4 ∐ BS 4 / (identifications)

The Quillen complex There is an exact sequence in cohomology: 0 → H ∗ ( G, F 2 ) → H ∗ ( D 8 , F 2 ) ⊕ H ∗ ( S 4 , F 2 ) ⊕ H ∗ ( S 4 , F 2 ) → → H ∗ ( D 8 , F 2 ) ⊕ H ∗ ( D 8 , F 2 ) → 0 The normalizer decomposition for A 2 ( G ) is sharp .