Related papers: Partial GVZ-groups

Let $G$ be a finite group of odd order. We show that if $\chi$ is an irreducible primitive character of $G$ then for all primes $p$ dividing the order of $G$ there is a conjugacy class such that the $p-$part of $\chi(1)$ divides the size of…

An irreducible character of a finite group $G$ is called quasi $p$-Steinberg character for a prime $p$ if it takes a nonzero value on every $p$-regular element of $G$. In this article, we classify the quasi $p$-Steinberg characters of…

For any group G, let C(G) denote the intersection of the normal- izers of centralizers of all elements of G. Set C0 = 1. Define Ci+1(G)=Ci(G) = C(G=Ci(G)) for i ? 0. By C1(G) denote the terminal term of the ascending series. In this paper,…

Call a compact, connected, simple Lie group $G$ {\emph{adjoint simple}} if it has trivial center. Let $C\subset G$ be a nontrivial conjugacy class, $e\in G$ the identity element of $G$. We prove the existence of an $N\in\mathbb{N}$,…

We define the notion of a semicharacter of a group G : A function from the group to C*, whose restriction to any abelian subgroup is a homomorphism. We conjecture that for any finite group, the order of the group of semicharacters is…

A finite group $G$ is of central type (in the non-classical sense) if it admits a non-degenerate cohomology class $[c]\in H^2(G,\C^*)$ ($G$ acts trivially on $\C^*$). Groups of central type play a fundamental role in the classification of…

Let $G$ be a finite group, and let $\mathrm{Irr}(G)$ denote the set of irreducible complex characters of $G$. An element $x$ of $G$ is said to be vanishing, if for some $\chi$ in $\mathrm{Irr}(G)$, we have $\chi(x)=0$. Also the element $x$…

Gcharacter tables of a finite group G were defined before. These tables can be very useful to obtain certain structural information of a normal subgroup from the character table of G. We analyze certain structural properties of normal…

Let G be a finite group. A collection P={H1, ..., Hr} of subgroups of G, where r > 1, is said a non-trivial partition of G if every non-identity element of G belongs to one and only one Hi, for some 1 <=i<=r. We call a group G that does not…

For a finite group $G$ and an irreducible complex character $\chi$ of $G$, the codegree of $\chi$ is defined by $\textrm{cod}(\chi)=|G:\textrm{ker}(\chi)|/\chi(1)$, where $\textrm{ker}(\chi)$ is the kernel of $\chi$. In this paper, we show…

A group $G$ is said to have restricted centralizers if for every $x\in G$ the centralizer $C_G(x)$ either is finite or has finite index in $G$. Shalev showed that a profinite group with restricted centralizers is virtually abelian. Here we…

Given a finite group $G$ and an irreducible complex character $\chi$ of $G$, the codegree of $\chi$ is defined as the integer ${\rm cod}(\chi)=|G:\ker\chi|/\chi(1)$. It was conjectured by G. Qian in [13] that, for every element $g$ of $G$,…

Let $G=G(K)$ be a simple algebraic group defined over an algebraically closed field $K$ of characteristic $p>0$. A subgroup $X$ of $G$ is said to be $G$-completely reducible if, whenever it is contained in a parabolic subgroup of $G$, it is…

Let $G$ be a finite group, and let cs$(G)$ be the set of conjugacy class sizes of $G$. Recalling that an element $g$ of $G$ is called a \emph{vanishing element} if there exists an irreducible character of $G$ taking the value $0$ on $g$, we…

We show that any connected algebraic group $G$ over a field admits a nilpotent normal subgroup $Z_\infty(G)$ such that the quotient $G/Z_\infty(G)$ has trivial center. We construct $Z_\infty(G)$ as the final term of the transfinitely…

Let $G$ be a finite $p$-group and $\chi,\psi$ be irreducible characters of $G$. We study the character $\chi\psi$ when $\chi\psi$ has at most $p-1$ distinct irreducible constituents.

In representation theory of finite groups an important role is played by irreducible characters of p-defect 0, for a prime p dividing the group order. These are exactly those vanishing at the p-singular elements. In this paper we generalize…

A cover of a finite non-cyclic group $G$ is a family $\mathcal{H}$ of proper subgroups of $G$ whose union equals $G$. A cover of $G$ is called minimal if it has minimal size, and irredundant if it does not properly contain any other cover.…

Let $G$ be a finite group and \( M \) be a maximal subgroup of \( G \). We call every irreducible constituent \( \chi \) of \( (1_M)^G \) a \( \mathcal{P} \)-character of \( G \) with respect to \( M \). In this paper, we prove that all…

The study of finite subgroups of a simple algebraic group $G$ reduces in a sense to those which are almost simple. If an almost simple subgroup of $G$ has a socle which is not isomorphic to a group of Lie type in the underlying…