Related papers: The simple $\mathscr{B}_{\psi}$-groups

An element $x$ in a finite group $G$ is said to be \textit{vanishing} if some (complex) irreducible character of $G$ takes value $0$ at $x$. In this article, we prove that every non-abelian finite simple group, except $\mathrm{SL}_2(4)$ and…

In recent years there has been significant progress in the study of products of subsets of finite groups and of finite simple groups in particular. In this paper we consider which families of finite simple groups $G$ have the property that…

The spectrum of a finite group is a set of its element orders. We prove that if $m>5$ then the group $L_{2^m}(2)\times L_{2^m}(2)\times L_{2^m}(2)$ is uniquely determined by its spectrum in the class of finite groups

For a finite group $G$, let $\sigma(G)$ be the number of subgroups of $G$ and $\sigma_\iota(G)$ the number of isomorphism types of subgroups of $G$. Let $L=L_r(p^e)$ denote a simple group of Lie type, rank $r$, over a field of order $p^e$…

A finite group $G$ is called an F-group if for every $x, y \in G \setminus Z(G)$, $C(x) \leq C(y)$ implies that $C(x) = C(y)$. On the otherhand, two elements of a group are said to be $z$-equivalent or in the same $z$-class if their…

The proper subgroup $B$ of the group $G$ is called {\it strongly embedded}, if $2\in\pi(B)$ and $2\notin\pi(B \cap B^g)$ for any element $g \in G \setminus B $ and, therefore, $ N_G(X) \leq B$ for any 2-subgroup $ X \leq B $. An element $a$…

A subgroup $H$ of a finite group $G$ is said to satisfy $\Pi$-property in $G$ if for every chief factor $L/K$ of $G$, $|G/K:N_{G/K}(HK/K\cap L/K)|$ is a $\pi(HK/K\cap L/K)$-number. A subgroup $H$ of $G$ is called to be $\Pi$-supplemented in…

Let $G$ be a finite group. Denote by $\textrm{Irr}(G)$ the set of all irreducible complex characters of $G.$ Let $\textrm{cd}(G)=\{\chi(1)\;|\;\chi\in \textrm{Irr}(G)\}$ be the set of all irreducible complex character degrees of $G$…

A supercharacter theory for a finite group $G$ is a set of superclasses each of which is a union of conjugacy classes together with a set of sums of irreducible characters called supercharacters that together satisfy certain compatibility…

T.C. Burness and S.D. Scott \cite{3} classified finite groups $G$ such that the number of prime order subgroups of $G$ is greater than $|G|/2-1$. In this note, we study finite groups $G$ whose subgroup graph contains a vertex of degree…

The spectrum $\omega(G)$ of a finite group $G$ is the set of element orders of $G$. If $\Omega$ is a non-empty subset of the set of natural numbers, $h(\Omega)$ stands for the number of isomorphism classes of finite groups $G$ with…

A remarkable result of Thompson states that a finite group is soluble if and only if its two-generated subgroups are soluble. This result has been generalized in numerous ways, and it is in the core of a wide area of research in the theory…

We refer to the set of the orders of elements of a finite group as its spectrum and say that finite groups are isospectral if their spectra coincide. In the paper we determine all finite groups isospectral to the simple groups $S_6(q)$,…

Let $G$ be a finite group and $S$ be a subset of $G.$ A bi-Cayley graph $\BCay(G,S)$ is a simple and an undirected graph with vertex-set $G\times\{1,2\}$ and edge-set $\{\{(g,1),(sg,2)\}\mid g\in G, s\in S\}$. A bi-Cayley graph $\BCay(G,S)$…

Let $G$ be a finite $p$-group of order $p^n$. It is known that $|\mathcal{M}(G)|=p^{\f{1}{2}n(n-1)-t(G)}$ and $t(G)\geq 0$. The structure of $G$ characterized when $t(G)\leq 4$ in \cite{be,el,ni,sa,zh}. The structure description of $G$ is…

Let G be a finite group, Irr(G) the set of all irreducible complex characters of G and X \in Irr(G). Let also cod(X) = |G : kerX|/X(1) and cod(G) = {cod(X) | X \in Irr(G)}. In this note, we show that the simple group PSL(2, q), for a prime…

In this paper, we provide new criteria for the solvability and supersolvability of a finite group based on its number of cyclic subgroups. A finite group G is called n-cyclic if it contains n cyclic subgroups. This paper also partially…

Suppose that $A$, $B$ and $S$ are non-empty subsets of a finite abelian group $G$. Then the generalized restricted sumset $$ A\stackrel{S}+B:=\{a+b:\,a\in A,\ b\in B,\ a-b\not\in S\} $$ contains at least $$ \min\{|A|+|B|-3|S|,p(G)\} $$…

Let $G$ be a finite group and $\chi\in \irr(G)$. The codegree of $\chi$ is defined as $\cod(\chi)=\frac{|G:\ker(\chi)|}{\chi(1)}$ and $\cod(G)=\{\cod(\chi) \ |\ \chi\in \irr(G)\}$ is called the set of codegrees of $G$. In this paper, we…

Let $\pi$ be a set of primes containing $2$ and an odd prime $p$. It is proved that if a finite group $G$ has a Hall $\pi$-subgroup $H$, then the non-$p$-soluble length of $G$ is bounded above by the generalized Fitting height of $H$. The…