Finite groups with the minimal generating set exchange property

Abstract

Let $d(G)$ be the smallest cardinality of a generating set of a finite group $G.$ We give a complete classification of the finite groups with the property that, whenever $⟨x_1,\dots ,x_{d(G)}⟩=⟨y_1,\dots ,y_{d(G)}⟩=G$, for any $1\le i\le d(G)$ there exists $1\le j\le d(G)$ such that $⟨x_1,\dots ,x_{i-1},y_j,x_{i+1},\dots ,x_{d(G)}⟩=G.$ We also prove that for every finite group $G$ and every maximal subgroup $M$ of $G$, there exists a generating set for $G$ of minimal size in which at least $d(G)-2$ elements belong to $M$. We conjecture that the stronger statement holds, that there exists a generating set of size $d(G)$ in which only one element does not belong to $M$, and we prove this conjecture for some suitable choices of $M$.