a subgroup $h$ is said to be $s$-permutable in a group $g$‎, ‎if‎ ‎$hp=ph$ holds for every sylow subgroup $p$ of $g$‎. ‎if there exists a‎ ‎subgroup $b$ of $g$ such that $hb=g$ and $h$ permutes with every‎ ‎sylow subgroup of $b$‎, ‎then $h$ is said to be $ss$-quasinormal in‎ ‎$g$‎. ‎in this paper‎, ‎we say that $h$ is a weakly $ss$-quasinormal‎ ‎subgroup of $g$‎, ‎if there is a normal subgroup ...

Abstract. A subgroup H of a group G is said to be SS-embedded in G if there exists a normal subgroup T of G such that HT is subnormal in G and H T H sG , where H sG is the maximal s- permutable subgroup of G contained in H. We say that a subgroup H is an SS-normal subgroup in G if there exists a normal subgroup T of G such that G = HT and H T H SS , where H SS is an SS-embedded subgroup of ...

a subgroup $h$ is said to be $s$-permutable in a group $g$‎, ‎if‎ ‎$hp=ph$ holds for every sylow subgroup $p$ of $g$‎. ‎if there exists a‎ ‎subgroup $b$ of $g$ such that $hb=g$ and $h$ permutes with every‎ ‎sylow subgroup of $b$‎, ‎then $h$ is said to be $ss$-quasinormal in‎ ‎$g$‎. ‎in this paper‎, ‎we say that $h$ is a weakly $ss$-quasinormal‎ ‎subgroup of $g$‎, ‎if there is a normal subgroup ...

Let G be a group hyperbolic relative to a collection of subgroups {Hλ, λ ∈ Λ}. We say that a subgroup Q ≤ G is hyperbolically embedded into G, if G is hyperbolic relative to {Hλ, λ ∈ Λ} ∪ {Q}. In this paper we obtain a characterization of hyperbolically embedded subgroups. In particular, we show that if an element g ∈ G has infinite order and is not conjugate to an element of some Hλ, λ ∈ Λ, th...

Elementary subgroups of relatively hyperbolic groups and bounded generation. Abstract Let G be a group hyperbolic relative to a collection of subgroups {H λ , λ ∈ Λ}. We say that a subgroup Q ≤ G is hyperbolically embedded into G, if G is hyperbolic relative to {H λ , λ ∈ Λ} ∪ {Q}. In this paper we obtain a characterization of hyperbolically embedded subgroups. In particular, we show that if an...

For a prime p, a proper subgroup H of the finite group G is strongly p-embedded in G if p divides |H| and p does not divide |H ∩ H| for all g ∈ G \H . A characteristic property of strongly p-embedded subgroups is that NG(X) ≤ H for any non-trivial p-subgroup X of H . Strongly p-embedded subgroups appear in the final stages of one of the programmes to better understand the classification of the ...