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

Let p be an arbitrary prime number and let P be a finite p-group. Let Ap(P) be the partially ordered set (poset for short) of all non-trivial elementary abelian subgroups of P ordered by inclusion and let Ap(P)⩾2 be the poset of all elementary abelian subgroups of P of rank at least 2. In [Serge Bouc, Jacques Thévenaz, The poset of elementary abelian subgroups of rank at least 2, Monogr. Enseign. Math. 40 (2008) 41–45], Bouc and Thévenaz proved that Ap(P)⩾2 has the homotopy type of a wedge of spheres (of possibly different dimensions). The general objective of this paper is to obtain more refined information on the homotopy type of the posets Ap(P) and Ap(P)⩾2. We give three different kinds of results in this direction.Firstly, we compute exactly the homotopy type of Ap(P)⩾2 when P is a p-group with a cyclic derived subgroup, that is we give the number of spheres occurring in each dimension in Ap(P)⩾2.Secondly, we compute a sharp upper bound on the dimension of the spheres occurring in Ap(P)⩾2 and give information on the p-groups for which this bound is reached.Thirdly, we determine explicitly for which of the p-groups with a cyclic derived subgroup the poset Ap(P) is homotopically Cohen–Macaulay.