Truncated Quillen coplexes of p-groups

Let G be a group and let p be a prime. Starting with the seminal papers of K. S. Brown and D. Quillen, relations between the algebraic structure of G and the topological structure of the order complex of the poset of its nontrivial elementary abelian p-subgroups have been studied. Say G is a finite p-group. Then this order complex is contractible and thus provides little information. However, S. Bouc and J. Th´evenaz showed that if one removes from the poset all subgroups of order p, the complex becomes more interesting. Here we show that determining the topology of this truncated complex is equivalent to counting certain extraspecial subgroups of G. This allows us to give a negative answer to a question raised by Bouc and Th´evenaz. (Received January 28, 2014)