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.

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.

A subgroup $H$ of a finite group $G$ is said to be conjugate-permutable if $H{H}^g=H^{g}H$ for all $g\in G$. More generaly, if we limit the element $g$ to a subgroup $R$ of $G$, then we say that the subgroup $H$ is $R$-conjugate-permutable. By means of the $R$-conjugate-permutable subgroups, we investigate the relationship between the nilpotence of $G$ and the $R$-conjugate-permutability of the Sylow subgroups of $A$ and $B$ under the condition that $G=AB$, where $A$ and $B$ are subgroups of $G$. Some results known in the literature are improved and...

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...

Let G be some metabelian 2-group satisfying the condition G/G’ ≃ ℤ/2ℤ × ℤ/2ℤ × ℤ/2ℤ. In this paper, we construct all the subgroups of G of index 2 or 4, we give the abelianization types of these subgroups and we compute the kernel of the transfer map. Then we apply these results to study the capitulation problem for the 2-ideal classes of some fields k satisfying the condition $Gal(k{₂}^{\left(2\right)}/k)\simeq G$, where $k{₂}^{\left(2\right)}$ is the second Hilbert 2-class field of k.

Let G be a finite group and p a prime. We consider an F-injector K of G, being F a Fitting class between Ep*p y Ep*Sp, and we study the structure and normality in G of the subgroups ZJ(K) and ZJ*(K), provided that G verifies certain conditions, extending some results of G. Glauberman (A characteristic subgroup of a p-stable group, Canad. J. Math.20(1968), 555-564).

We prove that if the average number of Sylow subgroups of a finite group is less than $\frac{41}{5}$ and not equal to $\frac{29}{4}$, then $G$ is solvable or $G/F\left(G\right)\cong A_5$. In particular, if the average number of Sylow subgroups of a finite group is $\frac{29}{4}$, then $G/N\cong A_5$, where $N$ is the largest normal solvable subgroup of $G$. This generalizes an earlier result by Moretó et al.