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On groups with few πβ²-character degrees
Dissertation (MSc)--University of Pretoria, 2022.
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| Format: | Thesis |
| Language: | English |
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University of Pretoria
2023
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| _version_ | 1867613438239309824 |
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| access_status_str | Open Access |
| author2 | Madanha, Sesuai Yash |
| author_browse | Madanha, Sesuai Yash |
| author_facet | Madanha, Sesuai Yash |
| collection | Thesis |
| dc_rights_str_mv | Β© 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. |
| description | Dissertation (MSc)--University of Pretoria, 2022. |
| format | Thesis |
| id | oai:repository.up.ac.za:2263/90274 |
| institution | University of Pretoria (South Africa) |
| language | English |
| last_indexed | 2026-06-10T12:36:08.960Z |
| license_str | Other β see source repository |
| provenance_str_mv | Harvested via OAI-PMH from UPSpace β University of Pretoria Institutional Repository |
| publishDate | 2023 |
| publishDateRange | 2023 |
| publishDateSort | 2023 |
| publisher | University of Pretoria |
| publisherStr | University of Pretoria |
| record_format | dspace |
| source_str | UPSpace β University of Pretoria Institutional Repository |
| spelling | oai:repository.up.ac.za:2263/90274 On groups with few πβ²-character degrees Madanha, Sesuai Yash shaunmabena@gmail.com Rodrigues, Bernardo Gabriel Mabena, Lehlogonolo Shaun p'-character degrees UCTD Character degrees Finite groups Irreducible characters Characters Dissertation (MSc)--University of Pretoria, 2022. Seitzβs 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 Seitzβs theorem is Thompsonβs celebrated theorem which states if the degrees of all non-linear irreducible characters of a group are divisible by a fixed prime π, then the group contains a normal π-complement. More recently, in 2020, as an extension to Thompsonβs theorem, Giannelli, Rizo, and Schaeffer Fry showed that if the character degree set of a group πΊ contains only two πβ²-character degrees (where π > 3 is a prime), then πΊ contains a normal subgroup π such that π has a normal π-complement and πΊ/π has a normal π-complement. Moreover, πΊ is solvable. In this dissertation, we explore a variation of Thompsonβs 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 π. We call such groups (β)-groups (π divides the order of the group). In 1998, Kazarin and Berkovich characterized the structure of (β)-groups. We give a detailed proof of their work for solvable groups. Moreover, we produce a classification of (β)-groups of order less than or equal to 100. DSI-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS) Mathematics and Applied Mathematics MSc Unrestricted 2023-03-30T10:17:49Z 2023-03-30T10:17:49Z 2023-03-04 2022 Dissertation * S2023 http://hdl.handle.net/2263/90274 en Β© 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. application/pdf University of Pretoria |
| spellingShingle | p'-character degrees UCTD Character degrees Finite groups Irreducible characters Characters On groups with few πβ²-character degrees |
| title | On groups with few πβ²-character degrees |
| title_full | On groups with few πβ²-character degrees |
| title_fullStr | On groups with few πβ²-character degrees |
| title_full_unstemmed | On groups with few πβ²-character degrees |
| title_short | On groups with few πβ²-character degrees |
| title_sort | on groups with few π character degrees |
| topic | p'-character degrees UCTD Character degrees Finite groups Irreducible characters Characters |
| url | http://hdl.handle.net/2263/90274 |