$m(G)-d(G)=1$的有限可解群$G$

On Finite Solvable Groups $G$ with $m(G)-d(G)=1$

中文关键词:  有限可解群  极小生成集  正规子群  循环群

英文关键词:finite solvable group  minimal generating set  normal subgroup  cyclic group

中文摘要:

      设$G$是有限群. $G$的生成集$X$称作是极小的如果$X$没有真子集生成$G$. 设$d(G)$和$m(G)$分别表示$G$的极小生成集所含元素的最小数和最大数.本文刻画了满足$m(G)-d(G)=1$, 且对任意非平凡正规子群$N$均有$m(G)\geq m(G/N)+m(N)$的有限可解群$G$.

英文摘要:

      Let $G$ be a finite group. A generating set $X$ of $G$ is said to be minimal if no proper subset of $X$ generates $G$. Let $d(G)$ and $m(G)$ denote the smallest size and the largest size of a minimal generating set of $G$, respectively. In this paper we present a characterization for finite solvable groups $G$ such that $m(G)-d(G)=1$ and $m(G)\geq m(G/N)+m(N)$ for any non-trivial normal subgroup $N$ of $G$.