Elementary abelian subgroups in some special p-groups

Abstract

Let $P$ be a finite $p$-group and $p$ be an odd prime. Let $\mathcal{A}_p(P)_{\geq2}$ be a poset consisting of elementary abelian subgroups of rank at least 2. If the derived subgroup $P'\cong C_p\times C_p$, then the spheres occurring in $\mathcal{A}_p(P)_{\geq2}$ all have the same dimension.