suppose that $h$ is a subgroup of $g$‎, ‎then $h$ is said to be‎ ‎$s$-permutable in $g$‎, ‎if $h$ permutes with every sylow subgroup of‎ ‎$g$‎. ‎if $hp=ph$ hold for every sylow subgroup $p$ of $g$ with $(|p|‎, ‎|h|)=1$)‎, ‎then $h$ is called an $s$-semipermutable subgroup of $g$‎. ‎in this paper‎, ‎we say that $h$ is partially $s$-embedded in $g$ if‎ ‎$g$ has a normal subgroup $t$ such that $ht...

Let G be a group and H subgroup of G. The notion being hyperbolically embedded in was introduced by Dahmani, Guiraldel, Osin. This note introduces an equivalent definition hyperbolic based on Bowditch’s approach to relatively groups terms fine graphs.

We introduce a new subgroup embedding property, namely S*-embedded and study the structure of finite group G under assumption that some subgroups prime order are in G. A serie known results unified extended.

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.