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.

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 $\mathrm{\Sigma }\left(G\right)$.

Theorem A yields the condition under which the number of solutions of equation $x^{p^k}=1$ in a finite $p$-group is divisible by $p^{n+k}$ (here $n$ is a fixed positive integer). Theorem B which is due to Avinoam Mann generalizes the counting part of the Sylow Theorem. We show in Theorems C and D that congruences for the number of cyclic subgroups of order $p^k$ which are true for abelian groups hold for more general finite groups (for example for groups with abelian Sylow $p$-subgroups).