Related papers: GVZ-groups with two character degrees

Let $G$ be a finite solvable group. We prove that if $\chi\in{\rm Irr}(G)$ has odd degree and $\chi(1)$ is the minimal degree of the non-linear irreducible characters of $G$, then $G/{\rm Ker} \chi$ is nilpotent-by-abelian.

We complete the description of group gradings on finite-dimensional incidence algebras. Moreover, we classify the finite-dimensional graded algebras that can be realized as incidence algebras endowed with a group grading.

we obtain a necessary condition for the character degree graph with all of its vertices are odd degree of a finite solvable group G.

Let G be a finite solvable group and $\chi\in \Irr(G)$ be a faithful character. We show that the derived length of G is bounded by a linear function of the number of distinct irreducible constituents of $\chi\bar{\chi}$. We also discuss…

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 $G$…

Marius T\u{a}rn\u{a}uceanu described the finite groups $G$ having $|G|-1$ cyclic subgroups. We describe the finite groups $G$ having $|G|-\Delta$ cyclic subgroups for $\Delta=2, 3, 4$ and $5$.

A finite group $G$ is a CG-group if $|{\rm Cent}(G) | = | G' |+2$, where $G'$ is the commutator subgroup and Cent$(G)$ is the set of distinct element centralizers of $G$. In this paper we give some results on CG-groups. We also give a…

Let G be a p-solvable group, P a p-subgroup and chi in Irr(G) such that chi(1)_p \ge |G:P|_p. We prove that the restriction chi_P is a sum of characters induced from subgroups Q\le P such that chi(1)_p=|G:Q|_p. This generalizes previous…

Let $q$ be a power of a prime $p$ and let $U(q)$ be a Sylow $p$-subgroup of a finite Chevalley group $G(q)$ defined over the field with $q$ elements. We first give a parametrization of the set $\text{Irr}(U(q))$ of irreducible characters of…

Let $\chi$ be a complex irreducible character of a finite group $G$. The conductor of $\chi$, denoted $c(\chi)$, is the smallest positive integer $n$ such that $\chi(x)\in \mathbb{Q}(\exp({2\pi i/n}))$ for all $x\in G$. We show that for…

In this paper, we obtain a characterization of GVZ-groups in terms of commutators and monolithic quotients. This characterization is based on counting formulas due to Gallagher.

Given a prime number $p$, every irreducible character $\chi$ of a finite group $G$ determines a unique conjugacy class of $p$-subgroups of $G$ which we will call the anchors of $\chi$. This invariant has been considered by L. Barker in the…

The character codegree of an irreducible character of a finite group $G$ is given by the index of its kernel in $G$ upon the character degree. We compute the codegrees of irreducible characters of VZ and Camina $p$-groups, and also obtain…

In this paper we characterize groups according to the number of end vertices in the associated coprime graphs. An upper bound on the order of the group that depends on the number of end vertices is obtained. We also prove that $2-$groups…

In this short note, it is proved that both the number of primitive characters and the number of quasi-primitive characters in a finite group $G$ is divisible by $|G:G'|$, where $G'$ is the derived subgroup of $G$.

If $G$ is a finite group, an irreducible complex-valued character $\chi$ is called rational if $\chi(g)$ is rational for all $g\in G$. Also, a conjugacy class $x^G$ is called rational, if for all irreducible complex-valued character $\chi$,…

We study finite p-groups G of coclass upto 4 for which the group Aut_z(G) of all central automorphisms of G is of minimal possible order. As a consequence, we obtain very short and elementary proofs of main results of Sharma and Gumber [7].

Let $G$ be a finite group. Denoting by ${\rm{cd}}(G)$ the set of the degrees of the irreducible complex characters of $G$, we consider the {\it character degree graph} of $G$: this is the (simple, undirected) graph whose vertices are the…

Let $\chi$ be an irreducible character of a group $G.$ We denote the sum of the codegrees of the irreducible characters of $G$ by $S_c(G)=\sum_{\chi\in {\rm Irr}(G)}{\rm cod}(\chi).$ We consider the question if $S_c(G)\leq S_c(C_n)$ is true…

An element $g$ of a finite group $G$ is said to be vanishing in $G$ if there exists an irreducible character $\chi$ of $G$ such that $\chi(g)=0$; in this case, $g$ is also called a zero of $G$. The aim of this paper is to obtain structural…