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

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          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.

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          Characters vanishing on all but two conjugacy classes

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            Groups Having Three Complex Irreducible Character Degrees

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              Sylow p-Subgroups of Solvable Camina Pairs

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                Author and article information

                Journal
                19 June 2012
                2012-10-08
                Article
                10.1515/jgt-2014-0011
                1206.4334
                5527f11d-6505-4dc9-8c10-b6dca2c7f759

                http://arxiv.org/licenses/nonexclusive-distrib/1.0/

                History
                Custom metadata
                20c15
                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 places
                math.GR

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