اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدExponent of a finite group of odd order with an involutory automorphism
133
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
Let $G$ be a finite group of odd order admitting an involutory automorphism $phi$. We obtain two results bounding the exponent of $[G,phi]$. Denote by $G_{-phi}$ the set ${[g,phi],vert, gin G}$ and by $G_{phi}$ the centralizer of $phi$, that is, the subgroup of fixed points of $phi$. The obtained results are as follows.1. Assume that the subgroup $langle x,yrangle$ has derived length at most $d$ and $x^e=1$ for every $x,yin G_{-phi}$. Suppose that $G_phi$ is nilpotent of class $c$. Then the exponent of $[G,phi]$ is $(c,d,e)$-bounded.2. Assume that $G_phi$ has rank $r$ and $x^e=1$ for each $xin G_{-phi}$. Then the exponent of $[G,phi]$ is $(e,r)$-bounded.
قيم البحث
اقرأ أيضاً
Let $G$ be a finite group admitting a coprime automorphism $phi$ of order $n$. Denote by $G_{phi}$ the centralizer of $phi$ in $G$ and by $G_{-phi}$ the set ${ x^{-1}x^{phi}; xin G}$. We prove the following results. 1. If every element from $G_{ph
i}cup G_{-phi}$ is contained in a $phi$-invariant subgroup of exponent dividing $e$, then the exponent of $G$ is $(e,n)$-bounded. 2. Suppose that $G_{phi}$ is nilpotent of class $c$. If $x^{e}=1$ for each $x in G_{-phi}$ and any two elements of $G_{-phi}$ are contained in a $phi$-invariant soluble subgroup of derived length $d$, then the exponent of $[G,phi]$ is bounded in terms of $c,d,e,n$.
In this article we present an extensive survey on the developments in the theory of non-abelian finite groups with abelian automorphism groups, and pose some problems and further research directions.
We study fixed point properties of the automorphism group of the universal Coxeter group Aut$(W_n)$. In particular, we prove that whenever Aut$(W_n)$ acts by isometries on complete $d$-dimensional CAT$(0)$ space with $d<lfloorfrac{n}{2}rfloor$, then
it must fix a point. We also prove that Aut$(W_n)$ does not have Kazhdans property (T). Further, strong restrictions are obtained on homomorphisms of Aut$(W_n)$ to groups that do not contain a copy of Sym(n).
We prove that the automorphism group of the braid group on four strands acts faithfully and geometrically on a CAT(0) 2-complex. This implies that the automorphism group of the free group of rank two acts faithfully and geometrically on a CAT(0) 2-co
mplex, in contrast to the situation for rank three and above.
A family $mathcal L$ of subsets of a set $X$ is called linked if $Acap B eemptyset$ for any $A,Binmathcal L$. A linked family $mathcal M$ of subsets of $X$ is maximal linked if $mathcal M$ coincides with each linked family $mathcal L$ on $X$ that con
tains $mathcal M$. The superextension $lambda(X)$ of $X$ consists of all maximal linked families on $X$. Any associative binary operation $* : Xtimes X to X$ can be extended to an associative binary operation $*: lambda(X)timeslambda(X)tolambda(X)$. In the paper we study automorphisms of the superextensions of finite monogenic semigroups and characteristic ideals in such semigroups. In particular, we describe the automorphism groups of the superextensions of finite monogenic semigroups of cardinality $leq 5$.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
