Bounding group orders by large character degrees: A question of Snyder

Abstract

Let $G$ be a nonabelian finite group and let $d$ be an irreducible character degree of $G$. Then there is a positive integer $e$ so that $|G| = d(d+e)$. Snyder has shown that if $e > 1$, then $|G|$ is bounded by a function of $e$. This bound has been improved by Isaacs and by Durfee and Jensen. In this paper, we will show for groups that have a nontrivial, abelian normal subgroup that $|G| \le e^4 - e^3$. We use this to prove that $|G| < e^4 + e^3$ for all groups. Given that there are a number of solvable groups that meet the first bound, it is best possible. Our work makes use of results regarding Camina pairs, Gagola characters, and Suzuki 2-groups.Comment: We added a proof in the case when p = 2 and G is nonsolvable. We also have fixed some mistakes, and improved the arguments in a couple of place

Picard groups for some blocks with TI defect groups

“…Now we need to study the structure of Aut(P ). We follow the notation in [16]. Identify P with the group A(m, Θ), where Θ is a non-trivial automorphism of F := GF (q) and elements of A(m, Θ) are given by pairs (a, b), a, b ∈ F and multiplication is defined by (a, c)(b, d) = (a + b, c + d + bΘ(a)).…”

On Finite Groups with an Irreducible Character Large Degree

“…Для e > 1 М. Айзекс в [5] показал, что |G| ≤ Be 6 для некоторой постоянной B. К. Дюрфи и С. Дженсен в [6] доказали, что |G| ≤ e 6 − e 4 . А по М. Льюису, d ≤ e 2 − e и |G| ≤ e 4 − e 3 -наилучшее возможное ограничение [7].…”

On generalized Gagola characters

“…Let G be a finite group, and denote by Irr(G) the set of irreducible complex characters of G. In the extreme case where G has a character χ ∈ Irr(G) so that χ vanishes on all but two conjugacy classes of G, Gagola [3] proved that such a character χ is unique when |G| > 2 and that G has a unique minimal normal subgroup N which is necessarily an elementary abelian p-group for some prime p. In this situation, following Lewis [9], we call χ a Gagola character of G and call (G, N ) a Gagola pair or a p-Gagola pair to emphasize the prime p when necessary.…”