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

Abstract

Let p be an arbitrary prime number and let P be a finite p-group.Let A p (P ) be the partially ordered set (poset for short) of all nontrivial elementary abelian subgroups of P ordered by inclusion and let A p (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 A p (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 A p (P ) and A p (P ) 2 . We give three different kinds of results in this direction. Firstly, we compute exactly the homotopy type of A p (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 A p (P ) 2 .Secondly, we compute a sharp upper bound on the dimension of the spheres occurring in A p (P ) 2 and give information on the pgroups for which this bound is reached. Thirdly, we determine explicitly for which of the p-groups with a cyclic derived subgroup the poset A p (P ) is homotopically CohenMacaulay.

Elementary abelian subgroups in some special 𝑝-groups

“…By [3,Proposition 4.7], we can see that the dimension of the sphere occurring in A p (P 1 ) ≥2 is l − 1 and the dimension of the sphere occurring in A p (P 2 ) ≥2 is s − 1. Since s = l, we can assume that s < l.…”

“…In [3], Bornand proved that Question 1.1 has a positive answer if P has a cyclic derived subgroup. Recently, [6] provided a negative answer to Question 1.1.…”

On transiso graph

“…Let G be a non-abelian finite p-group of order either p 4 or p5 . Then Φ(G) is not a cyclic group unless |Φ(G)| = p.Proof.…”