Abstract A group is called quasihamiltonian if all its subgroups are permutable, and we say that a subgroup Q of G permutably embedded in $\langle Q,g\rangle $ for each element g . It proved here contains normal such $G/Q$ ?ernikov, then has finite index; moreover, periodic, it ?ernikov N $G/N$ quasihamiltonian. This result should be compared with theorems Schlette stating over centre, abelian-...

Abstract It is known that if G a group such the centre factor $$G/\zeta (G)$$ G / ζ ( ) polycyclic, then also commutator subgroup $$G'$$ ′ polycyclic. The aim of this paper to des...

In this paper we study finite groups which possess a strongly pembedded subgroup for some prime p. Suppose that p is a prime. A subgroup H of the finite group G is said to be strongly p-embedded in G if the following two conditions hold. (i) H < G and p divides |H|; and (ii) if g ∈ G \H , then p does not divide |H ∩H|. One of the most important properties of strongly p-embedded subgroups is tha...

If P is a p-group for some prime p we shall write M (P ) to denote the set of all maximal subgroups of P and Md(P ) = {P1, ..., Pd} to denote any set of maximal subgroups of P such that ∩d i=1 Pi = Φ(P ) and d is as small as possible. In this paper, the structure of a finite group G under some assumptions on the c-normal or s-quasinormally embedded subgroups in Md(P ), for each prime p, and Syl...

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...

in this paper, we extend the construction of a fuzzy subgroup generated by a fuzzy subset to $l$-setting. this construction is illustrated by an example. we also prove that for an $l$-subset of a group, the subgroup generated by its level subset is the level subset of the subgroup generated by that $l$-subset provided the given $l$-subset possesses sup-property.

Abstract We give technical conditions for a quasi-isometry of pairs to preserve subgroup being hyperbolically embedded. consider applications the and commensurability invariance acylindrical hyperbolicity finitely generated groups.