Related papers: The binary actions of alternating groups

We continue our investigation of binary actions of simple groups. In this paper, we demonstrate a connection between the graph $\Gamma(\mathcal{C})$ based on the conjugacy class $\mathcal{C}$ of the group $G$, which was introduced in our…

Let $\mathcal{C}$ be a conjugacy class of involutions in a group $G$. We study the graph $\Gamma(\mathcal{C})$ whose vertices are elements of $\mathcal{C}$ with $g,h\in\mathcal{C}$ connected by an edge if and only if $gh\in\mathcal{C}$. For…

In this paper we introduce the graph $\Gamma_{sc}(G)$ associated with a group $G$, called the solvable conjugacy class graph (abbreviated as SCC-graph), whose vertices are the nontrivial conjugacy classes of $G$ and two distinct conjugacy…

Let $G$ be a finite group and let $N$ be a normal subgroup of $G$. We attach to $N$ two graphs ${\Gamma}_G(N)$ and ${\Gamma}^{\ast}_G(N)$ related to the conjugacy classes of $G$ contained in $N$ and to the set of primes dividing the sizes…

Let $A$ be a group acting by automorphisms on the group $G.$ \textit{The commuting graph $\Gamma(G,A)$ of $A$-orbits} of this action is the simple graph with vertex set $\{x^{A} : 1\ne x \in G \}$, the set of all $A$-orbits on $G\setminus…

Given a finite group $G$, denote by $\Gamma(G)$ the simple undirected graph whose vertices are the distinct sizes of noncentral conjugacy classes of $G$, and set two vertices of $\Gamma(G)$ to be adjacent if and only if they are not coprime…

Given a finite group $G$ with a normal subgroup $N$, the simple graph $\Gamma_\textit{G}( \textit{N} )$ is a graph whose vertices are of the form $|x^G|$, where $x\in{N\setminus{Z(G)}}$, and $x^G$ is the $G$-conjugacy class of $N$…

There are several graphs defined on groups. Among them we consider graphs whose vertex set consists conjugacy classes of a group $G$ and adjacency is defined by properties of the elements of conjugacy classes. In particular, we consider…

Let $G$ be $2$-generated group. The generating graph of $\Gamma(G)$ is the graph whose vertices are the elements of $G$ and where two vertices $g$ and $h$ are adjacent if $G=\langle g,h\rangle$. This graph encodes the combinatorial…

Let $\Gamma$ be a simple finite graph with vertex set $V(\Gamma)$ and edge set $E(\Gamma)$. Let $\mathcal{R}$ be an equivalence relation on $V(\Gamma)$. The $\mathcal{R}$-super $\Gamma$ graph $\Gamma^{\mathcal{R}}$ is a simple graph with…

Let $G$ be a finite group and $N(G)$ be the set of its conjugacy class sizes excluding~$1$. Let us define a directed graph $\Gamma(G)$, the set of vertices of this graph is $N(G)$ and the vertices $x$ and $y$ are connected by a directed…

Let $A$ be a finite group acting by automorphisms on the finite group $G$. We introduce the commuting graph $\Gamma (G,A)$ of this action and study some questions related to the structure of $G$ under certain graph theoretical conditions on…

A conjugacy class $C$ of a finite group $G$ is a sign conjugacy class if every irreducible character of $G$ takes value 0, 1 or -1 on $C$. In this paper we classify the sign conjugacy classes of alternating groups.

Let A_n be the alternating group on n letters. For n>5, we describe the elements alpha, beta in A_n when alpha^{A_n} beta^{A_n} is the union of at most four distinct conjugacy classes.

Let $G$ be a finite group and $N$ a normal subgroup of $G$. We determine the structure of $N$ when the graph $\Gamma_G(N)$, which is the graph associated to the conjugacy classes of $G$ contained in $N$, has no triangles and when the graph…

The {\it prime graph} $\Gamma(G)$ of a finite group $G$ is the graph whose vertex set is the set of prime divisors of $|G|$ and in which two distinct vertices $r$ and $s$ are adjacent if and only if there exists an element of $G$ of order…

We introduce the notion of $(G, \Gamma)$-crossed action on a tensor category, where $(G, \Gamma)$ is a matched pair of finite groups. A tensor category is called a $(G, \Gamma)$-crossed tensor category if it is endowed with a $(G,…

The commuting graph ${\Gamma(G)}$ of a group $G$ is the simple undirected graph with group elements as a vertex set and two elements $x$ and $y$ are adjacent if and only if $xy=yx$ in $G$. By eliminating the identity element of $G$ and all…

Given a finite group $G$, the generating graph $\Gamma(G)$ of $G$ has as vertices the (nontrivial) elements of $G$ and two vertices are adjacent if and only if they are distinct and generate $G$ as group elements. In this paper we…

Let $X$ be a non-empty set of positive integers and $X^*=X\setminus \{1\}$. The divisibility graph $D(X)$ has $X^*$ as the vertex set and there is an edge connecting $a$ and $b$ with $a, b\in X^*$ whenever $a$ divides $b$ or $b$ divides…