The groups of order p6 (p [more than or equal to] 3)

Abstract

Possibly the first thing that anyone asks for in any abstract algebraic system is a list of examples. In the case of groups, a complete list of all those groups G with a fixed number N of elements is still unknown for general values of N although the result has been known for many years in the case where N has only a few prime factors. Also a complete list of all finite abelian groups is well known. In the case where all the prime factors of N are the same, N = p" (p prime) and we call G ap - group. A lot more is known about the structure of p - groups than most other groups and so it is hoped that a complete list of these may be determined. In fact the groups of order p™ (n <5, p prime > 3) have been known since Schreier's listing of them in 1926. Also, in 1964, M. Hall and J. Senior published a list of the groups of order 2" (n < 6). In this thesis, the number of non-abelian groups of order p° (p prime > 3) is found to be "7, 13p* + 133p + 1346 + 80(p-1,3) + 45(p-1,4) + 8(p-1,5) + 8(p-1,6)} (where (p-1,n) is the greatest common divisor of p-1 and n) and the number of non-abelian groups of order - is found to be 481. The results of 0. Schreier have also been checked and extended to the case p = 3. This is done by considering in turn, families of groups which have the same commutator relations and isomorphic central factor groups (called "isoclinism families"). The remaining relations are determined for these groups checking the possibility that two sets of relations defin- isomorphic groups and exhibiting a canonical set of generators and relations for each isomorphism class. In chapter 2, the isoclinism families are determined and in chapter 3, the isomorphism classes for each family are determined in terms of corresponding equivalence classes of matrices over GF(p). The results of this work are contained in a list of the groups of order p° (p > 3) at the end of the thesis (except for one family which is still not determined).