Automorphism towers of general linear groups
| dc.contributor.author | Jónsdóttir, Margrét Sóley, author | |
| dc.contributor.author | Hulpke, Alexander, advisor | |
| dc.date.accessioned | 2024-03-13T19:53:53Z | |
| dc.date.available | 2024-03-13T19:53:53Z | |
| dc.date.issued | 2008 | |
| dc.description.abstract | Let G0, be a group, G 1 be the automorphism group of G0, G2 the automorphism group of G1 etc. The sequence of these groups together with the natural homomorphisms πi,i+1 : Gi → Gi+1, which take each element to the inner automorphism it induces, is called the automorphism tower of G 0. If πi,i+1 is an isomorphism for some i then the automorphism tower of G is said to terminate. For a given group it is in general not easy to say whether its automorphism tower terminates. Wielandt showed in 1939 that if G is finite with a trivial center then the automorphism tower of G will terminate in a finite number of steps. Since then, some sporadic examples of automorphism towers of finite groups have been described but no general results have been proven. In this thesis we study automorphism towers of finite groups with a non-trivial center. We look at the two extremes: (1) Groups which are center-rich. (2) Groups which have a small but non-trivial center. We show that when looking for an infinite family of groups with terminating automorphism towers the first case is unfeasible. We then turn our attention to the latter case, specifically general linear groups of dimension at least two. In odd characteristic GL(2, q) is not a split extension of the center. The first thing we do is to calculate the automorphism group of GL(2, q) for odd prime powers q. We provide explicit generators and describe the structure of Aut(GL(2, q)) in terms of well-known groups. In this case, the first automorphism group in the tower is a subdirect product of two characteristic factors. This structure is propagated through the tower and we use it to reduce the problem to studying subgroups of automorphism groups of smaller groups. We then use this structure to compute examples of automorphism towers of GL(2, q). | |
| dc.format.medium | born digital | |
| dc.format.medium | doctoral dissertations | |
| dc.identifier | ETDF_Jonsdottir_2008_3346429.pdf | |
| dc.identifier.uri | https://hdl.handle.net/10217/237802 | |
| dc.language | English | |
| dc.language.iso | eng | |
| dc.publisher | Colorado State University. Libraries | |
| dc.relation.ispartof | 2000-2019 | |
| dc.rights | Copyright and other restrictions may apply. User is responsible for compliance with all applicable laws. For information about copyright law, please see https://libguides.colostate.edu/copyright. | |
| dc.rights.license | Per the terms of a contractual agreement, all use of this item is limited to the non-commercial use of Colorado State University and its authorized users. | |
| dc.subject | automorphism towers | |
| dc.subject | general linear groups | |
| dc.subject | mathematics | |
| dc.title | Automorphism towers of general linear groups | |
| dc.type | Text | |
| dcterms.rights.dpla | This Item is protected by copyright and/or related rights (https://rightsstatements.org/vocab/InC/1.0/). You are free to use this Item in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you need to obtain permission from the rights-holder(s). | |
| thesis.degree.discipline | Mathematics | |
| thesis.degree.grantor | Colorado State University | |
| thesis.degree.level | Doctoral | |
| thesis.degree.name | Doctor of Philosophy (Ph.D.) |
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