Generators and minimal normal subgroups (Q1805509)

The author addresses the problem that given a minimal normal subgroup \(N\) in a finite group \(G\neq N\), compare the least number \(d(G)\) of elements required to generate \(G\) with \(d(G/N)\). It is proven here that \(d(G)\leq d(G/N)+1\). Furthermore if \(t\) is the length of a chief series in \(G\), then \(d(G)\leq t+1\). In addition, if \(m\) is the number of nonabelian factors in an arbitrary chief series, then the presentation rank of \(G\) is at most \(m\). A key point in the proof of the main theorem rests on a recently proven result by the author and \textit{F. Dalla Volta} that for a finite nonabelian simple group \(S\), for every pair \(\{x,y\}\) of elements in \(\text{Aut}(S)\), the subgroup \(\langle x,y,\text{Inn}(S)\rangle\) can be generated by two elements [J. Algebra 178, No. 1, 194-223 (1995)].