ترغب بنشر مسار تعليمي؟ اضغط هنا

On $sigma$-arithmetic graphs of finite groups

110   0   0.0 ( 0 )
 نشر من قبل Alexander Skiba
 تاريخ النشر 2020
  مجال البحث
والبحث باللغة English




اسأل ChatGPT حول البحث

Let $G$ be a finite group and $sigma$ a partition of the set of all? primes $Bbb{P}$, that is, $sigma ={sigma_i mid iin I }$, where $Bbb{P}=bigcup_{iin I} sigma_i$ and $sigma_icap sigma_j= emptyset $ for all $i e j$. If $n$ is an integer, we write $sigma(n)={sigma_i mid sigma_{i}cap pi (n) e emptyset }$ and $sigma (G)=sigma (|G|)$. We call a graph $Gamma$ with the set of all vertices $V(Gamma)=sigma (G)$ ($G e 1$) a $sigma$-arithmetic graph of $G$, and we associate with $G e 1$ the following three directed $sigma$-arithmetic graphs: (1) the $sigma$-Hawkes graph $Gamma_{Hsigma }(G)$ of $G$ is a $sigma$-arithmetic graph of $G$ in which $(sigma_i, sigma_j)in E(Gamma_{Hsigma }(G))$ if $sigma_jin sigma (G/F_{{sigma_i}}(G))$; (2) the $sigma$-Hall graph $Gamma_{sigma Hal}(G)$ of $G$ in which $(sigma_i, sigma_j)in E(Gamma_{sigma Hal}(G))$ if for some Hall $sigma_i$-subgroup $H$ of $G$ we have $sigma_jin sigma (N_{G}(H)/HC_{G}(H))$; (3) the $sigma$-Vasilev-Murashko graph $Gamma_{{mathfrak{N}_sigma }}(G)$ of $G$ in which $(sigma_i, sigma_j)in E(Gamma_{{mathfrak{N}_sigma}}(G))$ if for some ${mathfrak{N}_{sigma }}$-critical subgroup $H$ of $G$ we have $sigma_i in sigma (H)$ and $sigma_jin sigma (H/F_{{sigma_i}}(H))$. In this paper, we study the structure of $G$ depending on the properties of these three graphs of $G$.



قيم البحث

اقرأ أيضاً

Let $G$ be a finite group and $sigma ={sigma_{i} | iin I}$ some partition of the set of all primes $Bbb{P}$, that is, $sigma ={sigma_{i} | iin I }$, where $Bbb{P}=bigcup_{iin I} sigma_{i}$ and $sigma_{i}cap sigma_{j}= emptyset $ for all $i e j$. We s ay that $G$ is $sigma$-primary if $G$ is a $sigma _{i}$-group for some $i$. A subgroup $A$ of $G$ is said to be: ${sigma}$-subnormal in $G$ if there is a subgroup chain $A=A_{0} leq A_{1} leq cdots leq A_{n}=G$ such that either $A_{i-1}trianglelefteq A_{i}$ or $A_{i}/(A_{i-1})_{A_{i}}$ is $sigma$-primary for all $i=1, ldots, n$, modular in $G$ if the following conditions hold: (i) $langle X, A cap Z rangle=langle X, A rangle cap Z$ for all $X leq G, Z leq G$ such that $X leq Z$, and (ii) $langle A, Y cap Z rangle=langle A, Y rangle cap Z$ for all $Y leq G, Z leq G$ such that $A leq Z$. In this paper, a subgroup $A$ of $G$ is called $sigma$-quasinormal in $G$ if $L$ is modular and ${sigma}$-subnormal in $G$. We study $sigma$-quasinormal subgroups of $G$. In particular, we prove that if a subgroup $H$ of $G$ is $sigma$-quasinormal in $G$, then for every chief factor $H/K$ of $G$ between $H^{G}$ and $H_{G}$ the semidirect product $(H/K)rtimes (G/C_{G}(H/K))$ is $sigma$-primary.
Throughout this paper, all groups are finite. Let $sigma ={sigma_{i} | iin I }$ be some partition of the set of all primes $Bbb{P}$. If $n$ is an integer, the symbol $sigma (n)$ denotes the set ${sigma_{i} |sigma_{i}cap pi (n) e emptyset }$. Th e integers $n$ and $m$ are called $sigma$-coprime if $sigma (n)cap sigma (m)=emptyset$. Let $t > 1$ be a natural number and let $mathfrak{F}$ be a class of groups. Then we say that $mathfrak{F}$ is $Sigma_{t}^{sigma}$-closed provided $mathfrak{F}$ contains each group $G$ with subgroups $A_{1}, ldots , A_{t}in mathfrak{F}$ whose indices $|G:A_{1}|$, $ldots$, $|G:A_{t}|$ are pairwise $sigma$-coprime. In this paper, we study $Sigma_{t}^{sigma}$-closed classes of finite groups.
Let $sigma ={sigma_{i} | iin I}$ be a partition of the set of all primes $Bbb{P}$ and $G$ a finite group. Let $sigma (G)={sigma _{i} : sigma _{i}cap pi (G) e emptyset$. A set ${cal H}$ of subgroups of $G$ is said to be a complete Hall $sigma $-set of $G$ if every member $ e 1$ of ${cal H}$ is a Hall $sigma _{i}$-subgroup of $G$ for some $iin I$ and $cal H$ contains exactly one Hall $sigma _{i}$-subgroup of $G$ for every $i$ such that $sigma _{i}in sigma (G)$. We say that $G$ is $sigma$-full if $G$ possesses a complete Hall $sigma $-set. A complete Hall $sigma $-set $cal H$ of $G$ is said to be a $sigma$-basis of $G$ if every two subgroups $A, B incal H$ are permutable, that is, $AB=BA$. In this paper, we study properties of finite groups having a $sigma$-basis. In particular, we prove that if $G$ has a a $sigma$-basis, then $G$ is generalized $sigma$-soluble, that is, $G$ has a complete Hall $sigma $-set and for every chief factor $H/K$ of $G$ we have $|sigma (H/K)|leq 2$. Moreover, answering to Problem 8.28 in [A.N. Skiba, On some results in the theory of finite partially soluble groups, Commun. Math. Stat., 4(3) (2016), 281--309], we prove the following Theorem A. Suppose that $G$ is $sigma$-full. Then every complete Hall $sigma$-set of $G$ forms a $sigma$-basis of $G$ if and only if $G$ is generalized $sigma$-soluble and for the automorphism group $G/C_{G}(H/K)$, induced by $G$ on any its chief factor $H/K$, we have either $sigma (H/K)=sigma (G/C_{G}(H/K))$ or $sigma (H/K) ={sigma _{i}}$ and $G/C_{G}(H/K)$ is a $sigma _{i} cup sigma _{j}$-group for some $i e j$.
In this paper we study prime graphs of finite groups. The prime graph of a finite group $G$, also known as the Gruenberg-Kegel graph, is the graph with vertex set {primes dividing $|G|$} and an edge $p$-$q$ if and only if there exists an element of o rder $pq$ in $G$. In finite group theory, studying the prime graph of a group has been an important topic for the past almost half century. Only recently prime graphs of solvable groups have been characterized in graph theoretical terms only. In this paper, we continue this line of research and give complete characterizations of several classes of groups, including groups of square-free order, metanilpotent groups, groups of cube-free order, and, for any $nin mathbb{N}$, solvable groups of $n^text{th}$-power-free order. We also explore the prime graphs of groups whose composition factors are cyclic or $A_5$ and draw connections to a conjecture of Maslova. We then propose an algorithm that recovers the prime graph from a dual prime graph.
The Dehn function and its higher-dimensional generalizations measure the difficulty of filling a sphere in a space by a ball. In nonpositively curved spaces, one can construct fillings using geodesics, but fillings become more complicated in subsets of nonpositively curved spaces, such as lattices in symmetric spaces. In this paper, we prove sharp filling inequalities for (arithmetic) lattices in higher rank semisimple Lie groups. When $n$ is less than the rank of the associated symmetric space, we show that the $n$-dimensional filling volume function of the lattice grows at the same rate as that of the associated symmetric space, and when $n$ is equal to the rank, we show that the $n$-dimensional filling volume function grows exponentially. This broadly generalizes a theorem of Lubotzky-Mozes-Raghunathan on length distortion in lattices and confirms conjectures of Thurston, Gromov, and Bux-Wortman.
التعليقات
جاري جلب التعليقات جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا