Related papers: McKay graphs for alternating and classical groups

Isaacs has defined a character to be super monomial if every primitive character inducing it is linear. Isaacs has conjectured that if $G$ is an $M$-group with odd order, then every irreducible character is super monomial. We prove that the…

Let $G$ be a finite group. For a fixed element $g$ in $G$ and a given subgroup $H$ of $G$, the relative $g$-noncommuting graph of $G$ is a simple undirected graph whose vertex set is $G$ and two vertices $x$ and $y$ are adjacent if $x \in…

We show that every finitely generated group G with an element of order at least $(5rank(G))^{12}$ admits a locally finite directed Cayley graph with automorphism group equal to G. If moreover G is not generalized dihedral, then the above…

Let $H$ be a finite abelian (commutative) group of order $n \geq 2$, and $m >1$ be an integer. We define the $m$-graph of $H$, denoted by $m-G(H)$, as a simple undirected graph with vertex set $H$, and two distinct vertices, $a, b \in H$,…

A graph $G$ is said to be $\preceq$-ubiquitous, where $\preceq$ is the minor relation between graphs, if whenever $\Gamma$ is a graph with $nG \preceq \Gamma$ for all $n \in \mathbb{N}$, then one also has $\aleph_0 G \preceq \Gamma$, where…

Fix $c\in (0,1)$ and let $\Gamma$ be a $\lfloor c n\rfloor$-regular digraph on $n$ vertices drawn uniformly at random. We prove that when $n$ is large, the (non-symmetric) adjacency matrix $M$ of $\Gamma$ is invertible with high…

For a finite group $G$ and for a fixed positive integer $k$, $k\geq 2$, the $k$-power graph of $G$ is an undirected simple graph with vertex set $G$ in which two distinct vertices $x$ and $y$ are adjacent if and only if $x^k=y$ or $y^k=x$.…

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…

For a finite group $G,$ we investigate the direct graph $\Gamma(G),$ whose vertices are the non-hypercentral elements of $G$ and where there is an edge $x\mapsto y$ if and only if $[x,_ny]=1$ for some $n \in \mathbb N.$ We prove that…

We present simple graph-theoretic characterizations of Cayley graphs for left-cancellative monoids, groups, left-quasigroups and quasigroups. We show that these characterizations are effective for the end-regular graphs of finite degree.

We study word maps with constants on symmetric groups. Even though there are mixed identities of bounded length that are valid for all symmetric groups, we show that no such identities hold in a metric sense. Moreover, we prove that word…

Let G be a Chevalley group scheme of rank l. We show that the following holds for some absolute constant d>0 and two functions p_0=p_0(l) and C=C(l,p). Let p>p_0 be a prime number and let G_n:=G(\Z/p^n\Z) be the family of finite groups for…

A complex irreducible character of a finite group G is said to be p-constant, for some prime p dividing the order of G, if it takes constant value at the set of p-singular elements of G. In this paper we classify irreducible p-constant…

Let $\Delta\colon G \to \mathrm{GL}(n, K)$ be an absolutely irreducible representation of an arbitrary group $G$ over an arbitrary field $K$; let $\chi\colon G \to K\colon g \mapsto \mathrm{tr}(\Delta(g))$ be its character. In this paper,…

Suppose that $\Gamma$ is a non-empty connected graph, $\mathfrak{G}$ is the fundamental group of a graph of groups over $\Gamma$, and $\mathcal{C}$ is a root class of groups (the last means that $\mathcal{C}$ contains non-trivial groups and…

A finite group $G$ is called Cayley integral if all undirected Cayley graphs over $G$ are integral, i.e., all eigenvalues of the graphs are integers. The Cayley integral groups have been determined by Kloster and Sander in the abelian case,…

A recurring theme in finite group theory is understanding how the structure of a finite group is determined by the arithmetic properties of group invariants. There are results in the literature determining the structure of finite groups…

Given a finite group $G$, the Engel graph of $G$ is a directed graph $\Gamma(G)$ encoding pairs of elements satisfying some Engel word. Namely, $\Gamma(G)$ is the directed graph, where the vertices are the non-hypercentral elements of $G$…

Let $G$ be a finite group, let $\pi(G)$ be the set of prime divisors of $|G|$ and let $\Gamma(G)$ be the prime graph of $G$. This graph has vertex set $\pi(G)$, and two vertices $r$ and $s$ are adjacent if and only if $G$ contains an…

A graph $\Gamma$ is said to be a semi-Cayley graph over a group $G$ if it admits $G$ as a semiregular automorphism group with two orbits of equal size. We say that $\Gamma$ is normal if $G$ is a normal subgroup of ${\rm Aut}(\Gamma)$. We…