Elementary abelian subgroups in some special p-groups

TL;DR

This paper investigates the structure of elementary abelian subgroups of certain finite p-groups, revealing that in specific cases, the associated spheres all share the same dimension, highlighting a uniformity in their topological structure.

Contribution

It characterizes the dimensions of spheres in the poset of elementary abelian subgroups for p-groups with a specific derived subgroup structure.

Findings

All spheres in the poset have the same dimension when P' is isomorphic to C_p×C_p

Provides a topological uniformity result for elementary abelian subgroups in certain p-groups

Enhances understanding of subgroup poset topology in finite 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.