The Poset of Elementary Abelian Subgroups Of

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 [9], 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 (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 [4], [3], [2], [10]). Actually it appeared much before in problems related to the classification of finite simple groups (see Sections 1 and 10 of [7]).