A generalisation of supersoluble groups

Abstract:

A $p$-soluble group $G$ belongs to the class $F(n,p)$ whenever the ranks of the $p$-chief factors of $G$ divide $n$ and $G$ has order coprime to $n$. A group in $F(n,p)$ is characterised by the embedding of its maximal subgroups. Whenever ${N_1}$ and ${N_2}$ are normal subgroups of $G$, of coprime indices in $G$, which lie in $F(n,p)$, then $G$ lies in $F(n,p)$ also. $F(n)$ denotes the intersection, taken over all primes $p$, of the classes $F(n,p)$. Simple groups all of whose proper subgroups lie in $F(n)$ are determined. Given an integer $n > 2$, there exist an integer $m$ with the same prime divisors as $n$ and a soluble group $G$ such that $G$ lies in $F(m)$ but $G$ does not possess a Sylow tower. (We may take $m = n$ provided that $n$ is not a multiple of 1806.) Furthermore, when $n$ is odd, an example of a soluble group $G$, all of whose proper subgroups lie in $F(n)$ but $G$ has no Sylow tower, is given.