Finite groups having chain difference one (Q687668)

A maximal chain of subgroups of a group \(G\) is an ascending chain of subgroups \(1 = M_ 0 < M_ 1 < \cdots < M_ k = G\) where each \(M_ i\) is a maximal subgroup of \(M_{i+1}\). The length of such a chain is \(k\). The chain difference of a finite group \(G\), \(\text{cd}(G)\), is the length of a longest maximal chain minus the length of a shortest maximal chain. A classical result of K. Iwasawa is: \(\text{cd}(G) = 0\) if and only if \(G\) is supersolvable. Using the classification of finite simple groups, this paper describes the structure of all finite groups \(G\) with \(\text{cd}(G) = 1\).