Related papers: Even degree characters in principal blocks

We classify finite-dimensional complex Hopf algebras $A$ which are pointed, that is, all of whose irreducible comodules are one-dimensional, and whose group of group-like elements $G(A)$ is abelian such that all prime divisors of the order…

We show that if $p$ is a prime and $G$ is a finite $p$-solvable group satisfying the condition that a prime $q$ divides the degree of no irreducible $p$-Brauer character of $G$, then the normalizer of some Sylow $q$-subgroup of $G$ meets…

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…

We classify the finite groups $G$ which satisfies the condition that every complex irreducible character,whose degree's square doesn't divide the index of its kernel in $G$, lies in the same Galois conjugacy class.

Let $G$ be a finite group and $p\in \pi(G)$, and let Irr$(G)$ be the set of all irreducible complex characters of $G$. Let $\chi \in {\rm Irr}(G)$, we write ${\rm cod}(\chi)=|G:{\rm ker} \chi|/\chi(1)$, and called it the codegree of the…

Let $G$ be a finite group and $p$ be a prime number dividing the order of $G$. An irreducible character $\chi$ of $G$ is called a quasi $p$-Steinberg character if $\chi(g)$ is nonzero for every $p$-regular element $g$ in $G$. In this paper,…

Let $n$ and $k$ be natural numbers such that $2^k < n$. We study the restriction to $\mathfrak{S}_{n-2^k}$ of odd-degree irreducible characters of the symmetric group $\mathfrak{S}_n$. This analysis completes the study begun in [Ayyer A.,…

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…

Given an odd prime $p$, we investigate the position of simple modules in the stable Auslander-Reiten quiver of the principal block of a finite group with non-cyclic abelian Sylow $p$-subgroups. In particular, we prove a reduction to finite…

We study here the graphs with seven vertices in an effort to classify which of them appear as the prime character degree graphs of finite solvable groups. This classification is complete for the disconnected graphs. Of the 853…

Let G be a finite group and N be a non-trivial normal subgroup of G, such that the average character degree of irreducible characters in Irr(G|N) is less than or equal to 16=5. Then we prove that N is solvable. Also, we prove the…

Let $p$ be a prime such that $p \geq 5$. Let $G$ be a finite $p$-solvable group and let $p^a$ be the largest power of $p$ dividing $\chi(1)$ for an irreducible character $\chi$ of $G$, we show that $|G:F(G)|_p \leq p^{5.5a}$. Let $G$ be a…

Let $G$ be a finite group and ${\rm cd}(G)$ denote the set of complex irreducible character degrees of $G$. In this paper, we prove that if $G$ is a finite group and $H$ is an almost simple group with socle $H_{0}= \, ^{2}{\rm G}_{2}(q)$,…

The classification of abelian groups of central type is well known. However, the description of non-abelian groups of central type which are known to be solvable, is far from being understood. In this paper we classify all groups of central…

Let $G$ be a finite group, $\Bbb{F}$ be one of the fields $\mathbb{Q},\mathbb{R}$ or $\mathbb{C}$, and $N$ be a non-trivial normal subgroup of $G$. Let ${\rm acd}_{\Bbb{F}}^{*}(G)$ and ${\rm acd}_{\Bbb{F},even}(G|N)$ be the average degree…

The minimal faithful permutation degree $\mu(G)$ of a finite group $G$ is the least integer $n$ such that $G$ is isomorphic to a subgroup of the symmetric group $S_n$. If $G$ has a normal subgroup $N$ such that $\mu(G/N) > \mu(G)$, then $G$…

For a finite nonabelian group $G$ let $\rat(G)$ be the largest ratio of degrees of two nonlinear irreducible characters of $G$. We show that nonabelian composition factors of $G$ are controlled by $\rat(G)$ in some sense. Specifically, if…

In this note we provide some counterexamples for the conjecture of Moret\'{o} on finite simple groups, which says that any finite simple group $G$ can determined in terms of its order $|G|$ and the number of elements of order $p$, where $p$…

Motivated by a connection between semi-regular relative difference sets and mutually unbiased bases, we study relative difference sets with parameters $(m,n,m,m/n)$ in groups of non-prime-power orders. Let $p$ be an odd prime. We prove that…

We classify the finite groups whose non-linear irreducible characters that are not conjugate under the natural Galois action have distinct degrees, therefore extending the results in Berkovich et al. [Proc. Amer. Math. Soc. {\bf 115}…