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Register a new userThe Prime Graphs of Some Classes of Finite Groups
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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 order $pq$ 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.
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In this paper we have investigated some properties of the power graph and commuting graph associated with a finite group, using their tree-numbers. Among other things, it has been shown that the simple group $L_2(7)$ can be characterized through the tree-number of its power graph. Moreover, the classification of groups with power-free decomposition is presented. Finally, we have obtained an explicit formula concerning the tree-number of commuting graphs associated with the Suzuki simple groups.
In various classes of infinite groups, we identify groups that are presentable by products, i.e. groups having finite index subgroups which are quotients of products of two commuting infinite subgroups. The classes we discuss here include groups of small virtual cohomological dimension and irreducible Zariski dense subgroups of appropriate algebraic groups. This leads to applications to groups of positive deficiency, to fundamental groups of three-manifolds and to Coxeter groups. For finitely generated groups presentable by products we discuss the problem of whether the factors in a presentation by products may be chosen to be finitely generated.
Let ${frak F}$ be a class of group and $G$ a finite group. Then a set $Sigma $ of subgroups of $G$ is called a emph{$G$-covering subgroup system} for the class ${frak F}$ if $Gin {frak F}$ whenever $Sigma subseteq {frak F}$. We prove that: {sl If a set of subgroups $Sigma$ of $G$ contains at least one supplement to each maximal subgroup of every Sylow subgroup of $G$, then $Sigma$ is a $G$-covering subgroup system for the classes of all $sigma$-soluble and all $sigma$-nilpotent groups, and for the class of all $sigma$-soluble $Psigma T$-groups.} This result gives positive answers to questions 19.87 and 19.88 from the Kourovka notebook.
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 $G$.
The power graph $mathcal{P}(G)$ of a finite group $G$ is the graph whose vertex set is $G$, and two elements in $G$ are adjacent if one of them is a power of the other. The purpose of this paper is twofold. First, we find the complexity of a clique--replaced graph and study some applications. Second, we derive some explicit formulas concerning the complexity $kappa(mathcal{P}(G))$ for various groups $G$ such as the cyclic group of order $n$, the simple groups $L_2(q)$, the extra--special $p$--groups of order $p^3$, the Frobenius groups, etc.
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