On p-nilpotence and solubility of groups

Abstract

Recall a result due to O. J. Schmidt that a finite group whose proper subgroups are nilpotent is soluble. The present note extends this result and shows that if all non-normal maximal subgroups of a finite group are nilpotent, then (i) it is soluble; (ii) it is p-nilpotent for some prime p; (iii) if it is not nilpotent, then the number of prime divisors contained in its order is between 2 and k + 2, where k is the number of normal maximal subgroups which are not nilpotent.Mathematics Subject Classification (2010). 20D20; 20F19.

On a class of generalised Schmidt groups

“…The results presented here spring from the classical results of Schmidt [13] about the structure of the minimal non-nilpotent groups and later developments from them ( [12], [2], [3], [9], [10]). Schmidt proved that if all the maximal subgroups of a group G are nilpotent, then G is soluble, and that, in addition, if G is not nilpotent, |G| has exactly two distinct prime factors, G has a normal Sylow subgroup and a cyclic non-normal Sylow subgroup.…”

“…The present paper furnishes extensions of the main results of Rose, Li and Guo, and was motivated by some ideas of the paper [11]. We consider families of non-nilpotent subgroups covering the non-nilpotent part of the group, and analyse how they determine the group structure.…”

“…In a recent paper [11], Li and Guo characterised Rose groups by means of certain families of normal non-nilpotent subgroups, and obtained more detailed information about the number of primes dividing the order of the group. They also gave an alternative proof for solubility.…”

Finite groups in which every maximal subgroup is nilpotent or normal or has $p'$-order

“…It is known that a group G is nilpotent or minimal non-nilpotent if every maximal subgroup of G is nilpotent, and a group G is nilpotent if every maximal subgroup of G is normal, see [5,Theorem 9.1.9] and [5,Theorem 5.2.4], respectively. As a generalization, combining the nilpotence and the normality of groups together, Li and Guo [3,Theorem 1.2] proved that if all non-normal maximal subgroups of a group G are nilpotent then G is solvable and G is p-nilpotent for some prime p, that is, G has a normal p-complement. It is clear that the hypothesis that all nonnormal maximal subgroups of a group G are nilpotent is equivalent to the hypothesis that every maximal subgroup of G is nilpotent or normal.…”