Groups with exactly one irreducible character of degree divisible by<i>p</i>

Daniel Goldstein, Robert M. Guralnick, Mark L. Lewis, Alexander Moretó, Gabriel Navarro, Huu Tiep Pham

Abstract

Let p be a prime.We characterize those finite groups which have precisely one irreducible character of degree divisible by p.Minimal situations constitute a classical theme in group theory.Not only do they arise naturally, but they also provide valuable hints in searching for general patterns.In this paper, we are concerned with character degrees.One of the key results on character degrees is the Itô-Michler theorem, which asserts that a prime p does not divide the degree of any complex irreducible character of a finite group G if and only if G has a normal, abelian Sylow p-subgroup.In [Isaacs et al. 2009], Isaacs together with the fourth, fifth, and sixth authors of this paper studied the finite groups that have only one character degree divisible by p.They proved, among other things, that the Sylow p-subgroups of those groups were metabelian.This suggested that the derived length of the Sylow p-subgroups might be related with the number of different character degrees divisible by p.However, nothing could be said in [Isaacs et al. 2009] on how large p-Sylow normalizers were inside G. (As a trivial example, the dihedral group of order 2n for n odd has a unique character degree divisible by 2, and a self-normalizing Sylow 2-subgroup of order 2.)In this paper, we go further and completely classify the finite groups with exactly one irreducible character of degree divisible by p.Our focus now therefore is not only on the set of character degrees but also on the multiplicity of the number of irreducible characters of each degree.In Section 1, we define the terms semiextraspecial, ultraspecial, and doubly transitive Frobenius groups of Dickson type.

Locations

• Algebra & Number Theory - View - PDF
• Project Euclid (Cornell University) - View - PDF