Guido’s Book of Conjectures

If p is a prime number, the poset of all nontrivial elementary abelian p subgroups of a finite group plays an important role in both group theory and representation theory. It was studied by Quillen [Qu], who proved among many other things that it is homotopy equivalent to the poset of all nontrivial p-subgroups. In the case of a p -group P , one might believe that this poset has no interest since it is contractible (because the poset of all p subgroups has a maximal element, namely P). However, it turns out that the subposet A(P)≥2 consisting of elementary abelian subgroups of rank at least 2 plays a key role in some recent work about endo-trivial and endo-permutation modules (see [CT], [BT], [Bo], [Th]). Actually it appeared much before in problems related to the classification of finite simple groups (see Sections 1 and 10 of [GLS]). The purpose of this note is to show the following result and then state an open question related to A(P)≥2 .