A note on influence of subgroup restrictions in finite group structure (Q581668)

The authors consider various group theoretic restrictions on certain subgroups of a finite group G and determine the structure of G in these circumstances. Among these restrictions, we quote the following ones: a) the indices of all nonnormal maximal subgroups of G are equal; outcome: G must be solvable. b) G is nonabelian and every minimal subgroup of G is selfcentralizing; outcome: the order of G is of the form \(p^ nq\), where \(p\neq q\) are primes. c) every minimal subgroup of G is complemented in G; G is supersolvable. In Proposition 1, the structure of G is determined provided that G has a maximal subgroup M whose intersection with any other maximal subgroup of G is trivial. This result was also obtained by \textit{G. Walls} [Math. Stud. 47(1985), 139-140 (1979; Zbl 0591.20026)].