Group enumeration
<p>The thesis centres around two problems in the enumeration of <em>p</em>-groups. Define <em>f</em><sub>φ</sub>(<em>p<sup>m</sup></em>) to be the number of (isomorphism classes of) groups of order <em>p<sup>m</sup>...
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| Format: | Thesis |
| Language: | English |
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1992
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| _version_ | 1797094866754207744 |
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| author | Blackburn, S Simon Robert Blackburn |
| author2 | Neumann, P |
| author_facet | Neumann, P Blackburn, S Simon Robert Blackburn |
| author_sort | Blackburn, S |
| collection | OXFORD |
| description | <p>The thesis centres around two problems in the enumeration of <em>p</em>-groups. Define <em>f</em><sub>φ</sub>(<em>p<sup>m</sup></em>) to be the number of (isomorphism classes of) groups of order <em>p<sup>m</sup></em> in an isoclinism class φ. We give bounds for this function as φ is fixed and <em>m</em> varies and as <em>m</em> is fixed and φ varies. In the course of obtaining these bounds, we prove the following result. We say a group is <em>reduced</em> if it has no non-trivial abelian direct factors. Then the rank of the centre <em>Z</em>(<em>P</em>) and the rank of the derived factor group <em>P|P'</em> of a reduced <em>p</em>-group <em>P</em> are bounded in terms of the orders of <em>P|Z</em>(<em>P</em>)<em>P'</em> and <em>P'</em>∩<em>Z</em>(<em>P</em>)</p>. <p>A long standing conjecture of Charles C. Sims states that the number of groups of order <em>p<sup>m</sup></em> is<br/> <em>p</em><sup><sup>2</sup>andfrasl;<sub>27</sub><em>m</em><sup>3</sup>+<em>O</em>(<em>m</em><sup>2</sup>)</sup>. (1)</p> <p>We show that the number of groups of nilpotency class at most 3 and order <em>p<sup>m</sup></em> satisfies (1). We prove a similar result concerning the number of graded Lie rings of order <em>p<sup>m</sup></em> generated by their first grading.</p> |
| first_indexed | 2024-03-07T04:19:53Z |
| format | Thesis |
| id | oxford-uuid:caac5ed0-44e3-4bec-a97e-59e11ea268af |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T04:19:53Z |
| publishDate | 1992 |
| record_format | dspace |
| spelling | oxford-uuid:caac5ed0-44e3-4bec-a97e-59e11ea268af2022-03-27T07:09:01ZGroup enumerationThesishttp://purl.org/coar/resource_type/c_db06uuid:caac5ed0-44e3-4bec-a97e-59e11ea268afGroup theoryAbelian p-groupsEnglishPolonsky Theses Digitisation Project1992Blackburn, SSimon Robert BlackburnNeumann, PNeumann, P<p>The thesis centres around two problems in the enumeration of <em>p</em>-groups. Define <em>f</em><sub>φ</sub>(<em>p<sup>m</sup></em>) to be the number of (isomorphism classes of) groups of order <em>p<sup>m</sup></em> in an isoclinism class φ. We give bounds for this function as φ is fixed and <em>m</em> varies and as <em>m</em> is fixed and φ varies. In the course of obtaining these bounds, we prove the following result. We say a group is <em>reduced</em> if it has no non-trivial abelian direct factors. Then the rank of the centre <em>Z</em>(<em>P</em>) and the rank of the derived factor group <em>P|P'</em> of a reduced <em>p</em>-group <em>P</em> are bounded in terms of the orders of <em>P|Z</em>(<em>P</em>)<em>P'</em> and <em>P'</em>∩<em>Z</em>(<em>P</em>)</p>. <p>A long standing conjecture of Charles C. Sims states that the number of groups of order <em>p<sup>m</sup></em> is<br/> <em>p</em><sup><sup>2</sup>andfrasl;<sub>27</sub><em>m</em><sup>3</sup>+<em>O</em>(<em>m</em><sup>2</sup>)</sup>. (1)</p> <p>We show that the number of groups of nilpotency class at most 3 and order <em>p<sup>m</sup></em> satisfies (1). We prove a similar result concerning the number of graded Lie rings of order <em>p<sup>m</sup></em> generated by their first grading.</p> |
| spellingShingle | Group theory Abelian p-groups Blackburn, S Simon Robert Blackburn Group enumeration |
| title | Group enumeration |
| title_full | Group enumeration |
| title_fullStr | Group enumeration |
| title_full_unstemmed | Group enumeration |
| title_short | Group enumeration |
| title_sort | group enumeration |
| topic | Group theory Abelian p-groups |
| work_keys_str_mv | AT blackburns groupenumeration AT simonrobertblackburn groupenumeration |