Determination of the Number of Non-Abelian Isomorphic Types of Certain Finite Groups

Abstract

The first part of this work established, with examples, the fact that there are more than one non-abelian isomorphic types of groups of order n = sp, (s,p) = 1, where s<p and p  1 (mod s) for 100 < p < 4000. The factors s and p are distinct primes. Specifically considered here are groups of order n = 2p, 3p, 5p, 7p, 11p and 13p. It was discovered that the number of non-abelian isomorphic types of groups of order n = sp, s<p increased as n increased. The defining relations of such non-abelian isomorphic groups were outlined and a scheme developed to generate the numbers for the non-abelian isomorphic types of such groups. The scheme helped in generating many examples of non-abelian isomorphic types of such groups. The situation where p  k (mod s), k > 1 was worked out and such groups have no non-abelian isomorphic types. This gave credence to the fact that a group of order 15 and its like do not have a non-abelian isomorphic type. It also generated the non-abelian isomorphic types of groups of order n = spq, where s, p and q are distinct primes considering the congruence relationships between the primes. It was seen that there are more non-abelian isomorphic types when q 1 (mod p), q  1 (mod s) and p  1 (mod s). When q is not congruent to 1 modulo p but congruent to 1 modulo s fewer non-abelian isomorphic types were obtained. Moreover, if q is not congruent to 1 modulo p, q not congruent to 1 modulo s, and p not congruent to 1 modulo s, there cannot be a non-abelian isomorphic type of a group of order n = spq. In this case groups of order n = 2pq, 3pq, 5pq and 7pq were considered. Later, proofs of the number of non-abelian isomorphic types for n =sp and n =spq using the examples earlier generated were given.