Related papers: Groups Having a Character of Maximal Degree

We characterize the finite groups of minimal order that admit an irreducible complex character of degree $p$ or $p^2$, where $p$ is a prime.

Let $N(n)$ denote the number of isomorphism types of groups of order $n$. We consider the integers $n$ that are products of at most $4$ not necessarily distinct primes and exhibit formulas for $N(n)$ for such $n$.

Let $G$ be a finite group and $Irr(G)$ the set of irreducible complex characters of $G$. Let $e_p(G)$ be the largest integer such that $p^{e_p(G)}$ divides $\chi(1)$ for some $\chi \in Irr(G)$. We show that $|G:\mathbf{F}(G)|_p \leq p^{k…

Let $p$ be a prime and let $G$ be a finite group such that the smallest prime that divides $|G|$ is $p$. We find sharp bounds, depending on $p$, for the commuting probability and the average character degree to guarantee that $G$ is…

We study the finite solvable groups $G$ in which every real element has prime power order. We divide our examination into two parts: the case $\textbf{O}_2(G)>1$ and the case $\textbf{O}_2(G)=1$. Specifically we proved that if…

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…

The generalized order $e_G(g)$ of an element $g$ of a group $G$ is the smallest positive integer $k$ such that there exist $x_1,\ldots,x_k \in G$ such that $g^{x_1} \ldots g^{x_k}=1$, where $g^x=x^{-1}gx$. Let $e(G) = \max \{e_G(g)\ |\ g…

Let $U_n$ denote the group of upper $n \times n$ unitriangular matrices over a fixed finite field $\mathbb{F}$ of order $q$. That is, $U_n$ consists of upper triangular $n \times n$ matrices having every diagonal entry equal to $1$. It is…

Let $\F_q$ be a field of characteristic $p$ with $q$ elements. It is known that the degrees of the irreducible characters of the Sylow $p$-subgroup of $GL_n(\F_q)$ are powers of $q$ by Issacs. On the other hand Sangroniz showed that this is…

There has been some interest on how the average character degree affects the structure of a finite group. We define, and denote by $ \mathrm{anz}(G) $, the average number of zeros of characters of a finite group $ G $ as the number of zeros…

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 $ \chi \in \mathrm{Irr}(G) $. Define $ \mathrm{cv}(G)=\{\chi(g)\mid \chi \in \mathrm{Irr}(G), g\in G \} $, $ \mathrm{cv}(\chi)=\{\chi(g)\mid g\in G \} $ and denote $ \mathrm{dl}(G) $ by the derived length of…

The {\it prime graph} $\Gamma(G)$ of a finite group $G$ is the graph whose vertex set is the set of prime divisors of $|G|$ and in which two distinct vertices $r$ and $s$ are adjacent if and only if there exists an element of $G$ of order…

The prime graph of a finite group $G$, which is denoted by ${\rm GK}(G)$, is a simple graph whose vertex set is comprised of the prime divisors of $|G|$ and two distinct prime divisors $p$ and $q$ are joined by an edge if and only if there…

For primes $p,e>2$ there are at least $p^{e-3}/e$ groups of order $p^{2e+2}$ that have equal multisets of isomorphism types of proper subgroups and proper quotient groups, isomorphic character tables, and power maps. This obstructs recent…

Let N be a minimal normal nonabelian subgroup of a finite group G. We will show that there exists a nontrivial irreducible character of N of degree at least 5 which is extendible to G. This result will be used to settle two open questions…

We obtain a necessary condition for the character degree graph of a solvable group G to be Eulerian.

We prove that if the average of the degrees of the irreducible characters of a finite group $G$ is less than 16/5, then $G$ is solvable. This solves a conjecture of I.M. Isaacs, M. Loukaki, and the first author. We discuss related…

For every finite quasisimple group of Lie type $G$, every irreducible character $\chi$ of $G$, and every element $g$ of $G$, we give an exponential upper bound for the character ratio $|\chi(g)|/\chi(1)$ with exponent linear in $\log_{|G|}…

Let $H$ be an abelian subgroup of a finite group $G$ and $\pi$ the set of prime divisors of $|H|$. We prove that $|H O_{\pi}(G)/ O_{\pi}(G)|$ is bounded above by the largest character degree of $G$. A similar result is obtained when $H$ is…