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Zeros of irreducible characters of finite groups
Thesis (PhD)--University of Pretoria, 2019.
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| Format: | Thesis |
| Language: | English |
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University of Pretoria
2019
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| _version_ | 1867613545526460417 |
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| access_status_str | Open Access |
| author2 | Tong-Viet, Hung |
| author_browse | Tong-Viet, Hung |
| author_facet | Tong-Viet, Hung |
| collection | Thesis |
| dc_rights_str_mv | © 2019 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. |
| description | Thesis (PhD)--University of Pretoria, 2019. |
| format | Thesis |
| id | oai:repository.up.ac.za:2263/70706 |
| institution | University of Pretoria (South Africa) |
| language | English |
| last_indexed | 2026-06-10T12:37:51.329Z |
| license_str | Other — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from UPSpace — University of Pretoria Institutional Repository |
| publishDate | 2019 |
| publishDateRange | 2019 |
| publishDateSort | 2019 |
| publisher | University of Pretoria |
| publisherStr | University of Pretoria |
| record_format | dspace |
| source_str | UPSpace — University of Pretoria Institutional Repository |
| spelling | oai:repository.up.ac.za:2263/70706 Zeros of irreducible characters of finite groups Tong-Viet, Hung madanhasy@gmail.com Van den Berg, John Le, Tung Madanha, Sesuai Yash UCTD Mathematics Thesis (PhD)--University of Pretoria, 2019. This work is a contribution to the classification of finite groups with an irreducible character that vanishes on exactly one conjugacy class. Specifically, in this thesis we study finite non-solvable groups G that satisfy the above property when the character is primitive. We show that G has a homomorphic image that is either an almost simple group or a Frobenius group. We then classify all finite non-solvable groups with a faithful primitive irreducible character that vanishes on one conjugacy class. Our results answer two questions of Dixon and Rahnamai Barghi, one partially and the other completely. A classical theorem of Burnside states that every irreducible character whose character degree is divisible by a prime number vanishes on at least one conjugacy class. Our results imply that if the degree of a primitive irreducible character of a finite group is divisible by two distinct primes, then the character vanishes on at least two conjugacy classes except when the group has a composition factor isomorphic to the Suzuki group Suz(8). This is an extension of Burnside's Theorem. Motivated by our result above, we show that for M-groups, groups of odd order and groups of derived length at most 3, if the character degree of an irreducible character of a group is divisible by two distinct primes, then the irreducible character vanishes on at least two conjugacy classes. For nilpotent groups, metabelian groups and groups whose distinct character degrees are pairwise relatively prime, we show that if the character degree of an irreducible character of a group is divisible by n distinct primes, then the irreducible character vanishes on at least n conjugacy classes for any positive integer n. This also holds when the group is solvable and the irreducible character is primitive. DST-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS) Mathematics and Applied Mathematics PhD Unrestricted 2019-07-15T10:40:05Z 2019-07-15T10:40:05Z 2019-09-05 2019 Thesis Madanha, SY 2019, Zeros of irreducible characters of finite groups, PhD Thesis, University of Pretoria, Pretoria, viewed yymmdd <http://hdl.handle.net/2263/70706> S2019 http://hdl.handle.net/2263/70706 en © 2019 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. application/pdf University of Pretoria |
| spellingShingle | UCTD Mathematics Zeros of irreducible characters of finite groups |
| title | Zeros of irreducible characters of finite groups |
| title_full | Zeros of irreducible characters of finite groups |
| title_fullStr | Zeros of irreducible characters of finite groups |
| title_full_unstemmed | Zeros of irreducible characters of finite groups |
| title_short | Zeros of irreducible characters of finite groups |
| title_sort | zeros of irreducible characters of finite groups |
| topic | UCTD Mathematics |
| url | http://hdl.handle.net/2263/70706 |