Group enumeration

Thesis



The thesis centres around two problems in the enumeration of p-groups. Define f_φ(p^m) to be the number of (isomorphism classes of) groups of order p^m 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^m is
p^{2andfrasl;₂₇m3+O(m2)}. (1)

We show that the number of groups of nilpotency class at most 3 and order p^m satisfies (1). We prove a similar result concerning the number of graded Lie rings of order p^m generated by their first grading.

Authors: Blackburn, S; Simon Robert Blackburn

• Publication date: 1992
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Subjects: Group theory; Abelian p-groups