Let $G$ be a finite group. A subgroup $H$ is called $S$-semipermutable in if $HG_p$ = $G_pH$ for any $G_p\in Syl_p(G)$ with $(|H|, p) 1$, where $p$ prime number divisible $|G|$. Furthermore, said to $NH$-embedded there exists normal $T$ of such that $HT$ Hall and $H \cap T \leq H_{\overline{s}G}$, $H_{\overline{s}G}$ the largest contained $H$, $SS$-quasinormal provided supplement $B$ permutes e...

0. Introduction. Let Qp be the field of p-adic numbers, and let Q∞ = R. Let Gp be a connected semisimpleQp-algebraic group. The unipotent radical of a proper parabolic Qp-subgroup of Gp is called a horospherical subgroup. Two horospherical subgroups are called opposite if they are the unipotent radicals of two opposite parabolic subgroups. In [5] and [6], we studied discrete subgroups generated...

Let H be a subgroup of a finite group G. We say that H is SS-semipermutable in Gif H has a supplement K in G such that H permutes with every Sylow subgroup X of Kwith (jXj; jHj) = 1. In this paper, the Structure of SS-semipermutable subgroups, and finitegroups in which SS-semipermutability is a transitive relation are described. It is shown thata finite solvable group G is a PST-group if and on...

We use geometric techniques to investigate several examples of quasi-isometrically embedded subgroups of Thompson’s group F . Many of these are explored using the metric properties of the shift map φ in F . These subgroups have simple geometric but complicated algebraic descriptions. We present them to illustrate the intricate geometry of Thompson’s group F as well as the interplay between its ...