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...

A subgroup $H$ of a group $G$ is said to be complemented in $G$ if there exists a subgroup $K$ of $G$ such that $G=HK$ and $H\cap K=1$. In this paper we determine the structure of finite groups with some complemented primary subgroups, and obtain some new results about $p$-nilpotent groups.

The major aim of the present paper is to strengthen a nice result of Shemetkov and Skiba which gives some conditions under which every non-Frattini G-chief factor of a normal subgroup E of a finite group G is cyclic. As applications, some recent known results are generalized and unified.

The intention of this paper is to provide an elementary proof of the following known results: Let G be a finite group of the form G = AB. If A is abelian and B has a nilpotent subgroup of index at most 2, then G is soluble.

New criteria of existence and conjugacy of Hall subgroups of finite groups are given.

In this paper we prove the following results. Let π be a set of prime numbers and G a finite π-soluble group. Consider U, V ≤ G and $H\in {\mathrm{Hall}}_{\pi }\left(G\right)$ such that $H\cap V\in {\mathrm{Hall}}_{\pi }\left(V\right)$ and $1\ne H\cap U\in {\mathrm{Hall}}_{\pi }\left(U\right)$. Suppose also $H\cap U$ is a Hall π-sub-group of some S-permutable subgroup of G. Then $H\cap U\cap V\in {\mathrm{Hall}}_{\pi }(U\cap V)$ and $\langle H\cap U,H\cap V\rangle \in {\mathrm{Hall}}_{\pi }\left(\langle U\cap V\rangle \right)$. Therefore,the set of all S-permutably embedded subgroups of a soluble group G into which a given Hall system Σ reduces is a sublattice of the lattice of all Σ-permutable subgroups of G. Moreover any two subgroups of this sublattice of coprimeorders permute.

Multiplication groups of (finite) loops with commuting inner permutations are investigated. Special attention is paid to the normal closure of the abelian permutation group.

Let $ℱ$ be a saturated formation containing the class of supersolvable groups and let $G$ be a finite group. The following theorems are presented: (1) $G\in ℱ$ if and only if there is a normal subgroup $H$ such that $G/H\in ℱ$ and every maximal subgroup of all Sylow subgroups of $H$ is either $c$-normal or $S$-quasinormally embedded in $G$. (2) $G\in ℱ$ if and only if there is a normal subgroup $H$ such that $G/H\in ℱ$ and every maximal subgroup of all Sylow subgroups of $F^{*}\left(H\right)$, the generalized Fitting subgroup of $H$, is either $c$-normal or $S$-quasinormally...

A subgroup $H$ of a finite group $G$ is said to be $ss$-supplemented in $G$ if there exists a subgroup $K$ of $G$ such that $G=HK$ and $H\cap K$ is $s$-permutable in $K$. In this paper, we first give an example to show that the conjecture in A. A. Heliel’s paper (2014) has negative solutions. Next, we prove that a finite group $G$ is solvable if every subgroup of odd prime order of $G$ is $ss$-supplemented in $G$, and that $G$ is solvable if and only if every Sylow subgroup of odd order of $G$ is $ss$-supplemented in $G$. These results improve...