A finite solvable group G is called an X-group if the subnormal subgroups of G permute with all the system normalizers of G. It is our purpose here to determine some of the properties of X-groups. Subgroups and quotient groups of X-groups are X-groups. Let M and N be normal subgroups of a group G of relatively prime order. If G/M and G/N are X-groups, then G is also an X-group. Let the nilpotent residual L of G be abelian. Then G is an X-group if and only if G acts by conjugation on L as a group...

In this paper we consider finite loops whose inner mapping groups are nilpotent. We first consider the case where the inner mapping group $I\left(Q\right)$ of a loop $Q$ is the direct product of a dihedral group of order $8$ and an abelian group. Our second result deals with the case where $Q$ is a $2$-loop and $I\left(Q\right)$ is a nilpotent group whose nonabelian Sylow subgroups satisfy a special condition. In both cases it turns out that $Q$ is centrally nilpotent.

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.

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...

Let $\sigma =\{{\sigma }_i:i\in I\}$ be some partition of the set of all primes $\mathbb{P}$, $G$ be a finite group and $\sigma \left(G\right)=\{{\sigma }_i:{\sigma }_i\cap \pi \left(G\right)\ne \varnothing \}$. A set $\mathscr{H}$ of subgroups of $G$ is said to be a complete Hall $\sigma$-set of $G$ if every non-identity member of $\mathscr{H}$ is a Hall ${\sigma }_i$-subgroup of $G$ and $\mathscr{H}$ contains exactly one Hall ${\sigma }_i$-subgroup of $G$ for every ${\sigma }_i\in \sigma \left(G\right)$. $G$ is said to be $\sigma$-full if $G$ possesses a complete Hall $\sigma$-set. A subgroup $H$ of $G$ is $\sigma$-permutable in $G$ if $G$ possesses a complete Hall $\sigma$-set $\mathscr{H}$ such that $H{A}^x$= $A^{x}H$ for all $A\in \mathscr{H}$ and all $x\in G$. A subgroup $H$ of $G$ is $\sigma$-permutably embedded in $G$ if $H$ is $\sigma$-full...