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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.
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
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 s
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
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 $s
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--