The thesis centres around two problems in the enumeration of p-groups. Define f<sub>φ</sub>(p<sup>m</sup>) to be the number of (isomorphism classes of) groups of order p<sup>m</sup> in an isoclinism class φ. We give bounds for this function as φ is fixed and m varies and as m is fixed and φ varies. In the course of obtaining these bounds, we prove the following result. We say a group is reduced if it has no non-trivial abelian direct factors. Then the rank of the centre Z(P) and the rank of the derived factor group P|P' of a reduced p-group P are bounded in terms of the orders of P|Z(P)P' and P'∩Z(P). A long standing conjecture of Charles C. Sims states that the number of groups of order p<sup>m</sup> is<br/> p<sup><sup>2</sup>andfrasl;<sub>27</sub>m<sup>3</sup>+O(m<sup>2</sup>)</sup>. (1) We show that the number of groups of nilpotency class at most 3 and order p<sup>m</sup> satisfies (1). We prove a similar result concerning the number of graded Lie rings of order p<sup>m</sup> generated by their first grading.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:305312 |
| Date | January 1992 |
| Creators | Blackburn, Simon R. |
| Contributors | Neumann, P. M. |
| Publisher | University of Oxford |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://ora.ox.ac.uk/objects/uuid:caac5ed0-44e3-4bec-a97e-59e11ea268af |
Page generated in 0.1807 seconds