On the homotopy type of the Quillen complex of finite soluble groups

Abstract

We prove that the homotopy type of the Quillen complex of a finite soluble group at the prime p = 2 is that of a wedge of spheres of possible different dimensions.  2004 Elsevier Inc. All rights reserved.

Elementary abelian subgroups in p-groups with a cyclic derived subgroup

“…In [6], Fumagalli claimed that Question 1.1 has a positive answer if the group G is solvable. Unfortunately, his proof relies on a result that turns out to be false (see [2]), so that to our knowledge the following question seems to remain opened.…”

Counterexamples to a fiber theorem

“…In [4], Fumagalli studies such upper intervals and claims to have proved that if p is an odd prime and G is a solvable group, then the poset A p (G) is homotopy equivalent to a wedge of spheres [4,Theorem 21]. A crucial step in the proof, namely [4,Lemmas 19 and 20], relies heavily on a particular fiber theorem [4,Corollary 5], which unfortunately turns out to be false.…”

“…A crucial step in the proof, namely [4,Lemmas 19 and 20], relies heavily on a particular fiber theorem [4,Corollary 5], which unfortunately turns out to be false. The aim of this note is to provide two counterexamples to [4,Corollary 5]. The second one will also serve as a counterexample to the wedge decomposition formula obtained by Fumagalli in the proof of his Lemma 19.…”

“…If G is a p-group, for some prime p, we denote by Ω 1 (G) the subgroup of G generated by all elements of order p in G In his paper [4], Fumagalli introduces, for…”

“…Fumagalli's proof of Formula (1) makes use of Formula (2) below, which is the statement of Corollary 5 in Fumagalli's paper [4].…”