On groups with few 𝑝′-character degrees

Seitz’s theorem asserts that a finite group has exactly one non-linear irreducible character of degree greater than one if and only if the group is either an extraspecial 2-group or the group is isomorphic to a one-dimensional affine group over some field. An extension of Seitz’s theorem is Thompson’s celebrated theorem which states if the degrees of all non-linear irreducible characters of a group are divisible by a fixed prime 𝑝, then the group contains a normal 𝑝-complement. More recently, in 2020, as an extension to Thompson’s theorem, Giannelli, Rizo, and Schaeffer Fry showed that if the character degree set of a group 𝐺 contains only two 𝑝′-character degrees (where 𝑝 > 3 is a prime), then 𝐺 contains a normal subgroup 𝑁 such that 𝑁 has a normal 𝑝-complement and 𝐺/𝑁 has a normal 𝑝-complement. Moreover, 𝐺 is solvable. In this dissertation, we explore a variation of Thompson’s Theorem. We explore the structure of finite groups that have exactly one non-linear irreducible character whose degree is non-divisible by a fixed prime 𝑝. We call such groups (∗)-groups (𝑝 divides the order of the group). In 1998, Kazarin and Berkovich characterized the structure of (∗)-groups. We give a detailed proof of their work for solvable groups. Moreover, we produce a classification of (∗)-groups of order less than or equal to 100.

Grantor dc:publisher

University of Pretoria

Year dc:date.issued

2022

Advisor dc:contributor.advisor

• Madanha, Sesuai Yash

• p'-character degrees
• UCTD
• Character degrees
• Finite groups
• Irreducible characters
• Characters

Statement dc:rights

• © 2022 University of Pretoria. All rights reserved. The copyright in this work vests in the University of Pretoria. No part of this work may be reproduced or transmitted in any form or by any means, without the prior written permission of the University of Pretoria.

Language dc:language.iso

en

Dc Identifier Other

S2023

OAI identifier oai:identifier

oai:repository.up.ac.za:2263/90274