Sets of prime power order generators of finite groups

Abstract

A subset \(X\) of prime power order elements of a finite group \(G\) is called pp-independent if there is no proper subset \(Y\) of \(X\) such that \(\langle Y,\Phi(G) \rangle = \langle X,\Phi(G) \rangle\), where \(\Phi(G)\) is the Frattini subgroup of \(G\). A group \(G\) has property \(\mathcal{B}_{pp}\) if all pp-independent generating sets of \(G\) have the same size. \(G\) has the pp-basis exchange property if for any pp-independent generating sets \(B_1, B_2\) of \(G\) and \(x\in B_1\) there exists \(y\in B_2\) such that \((B_1\setminus \{x\})\cup \{y\}\) is a pp-independent generating set of \(G\). In this paper we describe all finite solvable groups with property \(\mathcal{B}_{pp}\) and all finite solvable groups with the pp-basis exchange property.