A bound on the order of a group having a large character degree

Abstract

Communicated by Michel BrouéKeywords: Irreducible character degree Group order upper bound Let d be the degree of an irreducible character of a finite group G.We can write |G| = d(d +e) for some non-negative integer e. In this document, we prove that if e > 1 then |G| < e 6 − e 4 . This improves an upper bound found by Isaacs of the form Be 6 , where B is an unknown universal constant. We also describe conditions sufficient to sharpen this bound to |G| e 4 − e 3 . In addition, we remove the appeal to the classification of simple groups, which is used in the original paper by Isaacs.

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

“…Proof. We may apply Theorem 5.2 of [5] to see that G has a character χ ∈ Irr(G) which dominates all the other irreducible characters of G and by Lemma 2.1 of [5], χ(1) = d. By Lemma 4.2 of [5], we know that G has a minimal normal subgroup N such that χ vanishes off of N and G acts transitively on N \ {1}. This implies that χ vanishes on all but two conjugacy classes of G, and so, χ is a Gagola character.…”

“…We first use the results in [5] to reduce the initial question to a question regarding Gagola pairs. Since |G| = d(d + e), to prove |G| ≤ e 4 − e 3 it suffices to prove that d ≤ e 2 − e and to prove |G| ≤ e 4 + e 3 , it suffices to prove that d ≤ e 2 .…”

“…Isaacs has shown that |G| ≤ Be 6 for some universal constant B and in many cases that |G| ≤ e 6 + e 4 (see [11]). Finally, Durfee and Jensen have proved in [5] (without using the classification of nonabelian simple groups) that |G| ≤ e 6 −e 4 . When G is solvable and either e is a prime or e is divisible by at least two distinct primes, they prove that |G| ≤ e 4 − e 3 .…”

“…Like [5], [11], and [20], we consider groups first studied by Gagola. Gagola studied groups that have an irreducible character that vanish on all but two conjugacy classes.…”

On Finite Groups with an Irreducible Character Large Degree

“…Я. Берковичем в [4] классифицированы группы при e = 1 и e = 2. 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].…”