Finite groups of chain difference one (Q1579157)

A finite group \(G\) is said to have chain difference one if the lengths of any two maximal chains of subgroups of \(G\) differ by at most one. Such groups were classified by \textit{B. Brewster, M. B. Ward} and \textit{I. Zimmermann} [J. Algebra 160, No. 1, 179-191 (1993; Zbl 0791.20016)], using the full Classification of Finite Simple Groups. The present article shows how one can use an elementary argument to reduce the problem of classifying finite groups with chain difference one to the study of certain finite simple groups having dihedral or semidihedral Sylow \(2\)-subgroups. Hence, they recover the result of Brewster et al. using only an elementary argument and the following two classification theorems by \textit{D. Gorenstein} and \textit{J. H. Walter} [J. Algebra 2, 85-151, 218-270, 354-393 (1965; Zbl 0192.11902, Zbl 0192.12001)], and \textit{J. L. Alperin, R. Brauer}, and \textit{D. Gorenstein} [Trans. Am. Math. Soc. 151, 1-261 (1970; Zbl 0222.20002)]. The question as to whether or not there exist infinitely many non-isomorphic finite simple groups with chain difference one is still open. The authors remark, however, that this question can be reformulated in purely number-theoretic terms.