On Finite solvable groups $G$ with $m(G)-d(G)=1$

Abstract:

      This paper is one of a series of papers that aim to give a characterization of finite groups $G$ with $m(G)-d(G)=1$, where $d(G)$ and $m(G)$ denote the minimal number of generators of $G$ and the largest size of a minimal generating set of $G$, respectively. In this paper we present such a characterization for finite solvable groups $G$ such that $m(G)\geq m(G/N)+m(N)$ for any non-trivial normal subgroup $N$ of $G$.