Abstract
Abstract For any given subgroup H of a finite group G, the Quillen poset ${\mathcal {A}}_p(G)$ of nontrivial elementary abelian p-subgroups is obtained from ${\mathcal {A}}_p(H)$ by attaching elements via their centralisers in H. We exploit this idea to study Quillen’s conjecture, which asserts that if ${\mathcal {A}}_p(G)$ is contractible then G has a nontrivial normal p-subgroup. We prove that the original conjecture is equivalent to the ${{\mathbb {Z}}}$ -acyclic version of the conjecture (obtained by replacing ‘contractible’ by ‘ ${{\mathbb {Z}}}$ -acyclic’). We also work with the ${\mathbb {Q}}$ -acyclic (strong) version of the conjecture, reducing its study to extensions of direct products of simple groups of p-rank at least $2$ . This allows us to extend results of Aschbacher and Smith and to establish the strong conjecture for groups of p-rank at most $4$ .
Highlights
Given a finite group and a prime number, let A ( ) be the Quillen poset of nontrivial elementary abelian -subgroups of
For a given subgroup ≤, we ‘inflate’ the subposet A ( ) and we show that the remaining points of A ( ) are attached to this inflated subposet throughout their centralisers in
We show that × gives a nonzero cycle in the homology of
Summary
Given a finite group and a prime number , let A ( ) be the Quillen poset of nontrivial elementary abelian -subgroups of. Aschbacher and Smith excluded the groups containing these components during their analysis of the conjecture for odd , mainly because the centralisers of their field automorphisms of order have nontrivial normal -subgroups (see section 5 for a more detailed discussion). Our theorem shows that we can suppose that does not contain these components if we are aiming to prove Quillen’s conjecture This allows us to extend the main result of Aschbacher and Smith to = 5. We combine Theorem 1 with the classification of the simple groups of low -rank, the structure of their centralisers and the classification of groups with a strongly -embedded subgroup (i.e., with disconnected Quillen’s complex) to yield the -rank 4 case of the conjecture. We include an appendix containing basic properties of the almost-simple groups with a strongly -embedded subgroup
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