Abstract A rigid automorphism of a linking system is an that restricts to the identity on Sylow subgroup. inner conjugation by element in center At odd primes, it known each centric inner. We prove group outer automorphisms at prime $2$ elementary abelian and splits over subgroup automorphisms. In second result, we show if finite G for , then $p'$ -order modulo automorphisms, provided has no no...

A subgroup of V containing a principal congruence subgroup T(«) is said to be a congruence subgroup, and is of level « if « is the least such integer. In a recent article [2] the writer determined all normal subgroups of T of genus 1 (see [l] for the definition of the genus of a subgroup of r). An interesting question that arises is to decide which of these are also congruence subgroups. In thi...

Abstract A subgroup functor τ \tau is said Φ \Phi -regular if for all primitive groups G G , whenever H ∈ ( ) H\in \left(G) a p p -subgroup and N N minimal normal of then <m:mo...

A measure preserving action of a countably in nite group Γ is called totally ergodic if every in nite subgroup of Γ acts ergodically. For example, all mixing and mildly mixing actions are totally ergodic. This note shows that if an action of Γ is totally ergodic then there exists a nite normal subgroup N of Γ such that the stabilizer of almost every point is equal to N. Surprisingly the proof r...

Abstract A group G is said to be an FC -group if every conjugacy class of has finite order and a T subnormal subgroup normal in . In this paper we give characterization groups that are both -groups go further the study with few non-normal subgroups sense chain conditions. The structure satisfying maximal, minimal double condition on will described.

An automorphism α of a group G is normal if it fixes every normal subgroup of G setwise. We give an algebraic description of normal automorphisms of relatively hyperbolic groups. In particular, we prove that for any relatively hyperbolic group G, Inn(G) has finite index in the subgroup Autn(G) of normal automorphisms. If, in addition, G is non-elementary and has no non-trivial finite normal sub...

The Pythagorean fuzzy set is an extension of the intuitionistic used to handle uncertain circumstances in various decisions making problems. Group theory a mathematical technique for dealing with problems symmetry. This study deals group theory. In this article, we characterize notion subgroup and examine algebraic properties concept. An extensive on cosets subgroup, normal subgroups performed....

Let Γ be a finitely generated residually finite group. Denote by sn(Γ) (resp. tn(Γ)) the number of subgroups (resp. normal subgroups) of Γ of index at most n. In the last two decades the study of the connection between the algebraic structure of Γ and the growth rate of the sequence {sn(Γ)}n=1 has become a very active area of research under the rubric “subgroup growth” (see [L1], [LS] and the r...

let $h$, $l$ and $x$ be subgroups of a finite group$g$. then $h$ is said to be $x$-permutable with $l$ if for some$xin x$ we have $al^{x}=l^{x}a$. we say that $h$ is emph{$x$-quasipermutable } (emph{$x_{s}$-quasipermutable}, respectively) in $g$ provided $g$ has a subgroup$b$ such that $g=n_{g}(h)b$ and $h$ $x$-permutes with $b$ and with all subgroups (with all sylowsubgroups, respectively) $v$...