Finite soluble groups satisfying the replacement property

Abstract

We investigate the finite soluble groups $G$ with the following property (replacement property): for every irredundant generating set $\{g_1,\dots ,g_m\}$ of maximal size and for any $1\ne g\in G$ there exists an $i\in \{1,\dots ,m\}$ so that $⟨g_1,\dots ,g_{i-1},g,g_{i+1},\dots ,g_m⟩=G.$