Groups Having a Character of Maximal Degree
Sara Jensen, Mark L. Lewis
TL;DR
The paper investigates Isaacs groups, i.e., finite groups with $|G|=e^4-e^3$ and a character degree $d=e^2-e$, proving that $e$ must be a prime power and developing a structural framework to analyze the equality case. It provides a thorough analysis of the $p$-closed case, where $G$ contains a normal Sylow $p$-subgroup $P$ with $(P,Z(P))$ a Camina pair and a two-transitive Frobenius action, and then treats non-$p$-closed scenarios via $O_p(G)$, Zsigmondy-type arguments, and Camina-pair structure. The authors classify Isaacs groups for $e$ prime and for $e=p^2$, describing explicit p-closed families and rare nonsolvable instances, including Camina-pair examples. They present extensive computational classifications for $e=4,9,25$, yielding several new solvable and nonsolvable instances and demonstrating the existence of Camina-pair structures beyond solvable examples. Degrees $49$ and $121$ are explored computationally to reveal ultraspecial-like patterns, guiding future classification and highlighting the complexity of the problem at larger prime powers. Overall, the work advances understanding of the equality case $|G|=e^4-e^3$, provides constructive methods to generate p-closed examples, and exhibits new nonsolvable Camina-pair phenomena within Isaacs groups.
Abstract
Let $G$ be a group, let $d$ be a character degree, and let $e$ be the integer so that $|G| = d(d+e)$. It has been shown when $e > 1$ that $|G| \le e^4 - e^3$. In this paper, we consider the groups where $|G| = e^4 - e^3$. It is known that $e$ must be a power of a prime. We classify the groups where $e$ is a prime and where $e$ is $4$, $9$, and $25$. In so doing, we find a new nonsolvable Camina pair.