It is well known that a group G = AB which is the product of two supersoluble subgroups A and B is not supersoluble in general. Under suitable permutability conditions on A and B, we show that for any minimal normal subgroup N both AN and BN are supersoluble. We then exploit this to establish some sufficient conditions for G to be supersoluble.

Gli autori studiano il sottogruppo $\mathrm{\Sigma }\left(G\right)$ intersezione dei sottogruppi massimali e non supersolubili di un gruppo finito $G$ e le relazioni tra la struttura di $G$ e quella di $\mathrm{\Sigma }\left(G\right)$.

In this paper, the structures of collection of pronormal subgroups of dicyclic, symmetric and alternating groups $G$ are studied in respect of formation of lattices $\mathrm{L}\left(G\right)$ and sublattices of $\mathrm{L}\left(G\right)$. It is proved that the collections of all pronormal subgroups of ${\mathrm{A}}_n$ and S${}_n$ do not form sublattices of respective $\mathrm{L}\left({\mathrm{A}}_n\right)$ and $\mathrm{L}\left({\mathrm{S}}_n\right)$, whereas the collection of all pronormal subgroups $\mathrm{LPrN}\left({\mathrm{Dic}}_n\right)$ of a dicyclic group is a sublattice of $\mathrm{L}\left({\mathrm{Dic}}_n\right)$. Furthermore, it is shown that $\mathrm{L}\left({\mathrm{Dic}}_n\right)$ and $\mathrm{LPrN}({\mathrm{Dic}}_n$) are lower semimodular lattices.

Suppose $G$ is a finite group and $H$ is a subgroup of $G$. $H$ is said to be $s$-permutably embedded in $G$ if for each prime $p$ dividing $\left|H\right|$, a Sylow $p$-subgroup of $H$ is also a Sylow $p$-subgroup of some $s$-permutable subgroup of $G$; $H$ is called weakly $s$-permutably embedded in $G$ if there are a subnormal subgroup $T$ of $G$ and an $s$-permutably embedded subgroup $H_{se}$ of $G$ contained in $H$ such that $G=HT$ and $H\cap T\le H_{se}$. We investigate the influence of weakly $s$-permutably embedded subgroups on the $p$-nilpotency and $p$-supersolvability of finite...