Finite groups whose maximal subgroups of order divisible by all the primes are supersolvable

TL;DR

This paper investigates finite groups where maximal subgroups divisible by all primes dividing the group are supersolvable, revealing the structure and limitations of such groups, including their composition factors and solvability properties.

Contribution

It characterizes the structure of these finite groups, showing nonabelian simple groups can appear as factors and establishing bounds on their nilpotency length and p-length.

Findings

Nonabelian simple groups can occur as composition factors.

Solvable groups can have arbitrarily large nilpotency length and rank.

The p-length of such groups is at most 1.

Abstract

We study finite groups $G$ with the property that for any subgroup $M$ maximal in $G$ whose order is divisible by all the prime divisors of $|G|$, $M$ is supersolvable. We show that any nonabelian simple group can occur as a composition factor of such a group and that, if $G$ is solvable, then the nilpotency length and the rank are arbitrarily large. On the other hand, for every prime $p$, the $p$-length of such a group is at most $1$. This answers questions proposed by V. Monakhov in The Kourovka Notebook.