A STUDY OF SOME PROPERTIES OF GENERALIZED GROUPS

Generalized group (G,•) consisting of a non empty set together with a binary operation which is associative, has unique identity and unique inverse for each element. The idea of subgroups, normal groups, factor groups and homomorphisms in the classical group are similar to those in generalized group. These notions were then employed to establish the following results: homomorphism theorems for generalized groups, factor generalized group G by generalized subgroup H, isomorphism theorem for generalized group, isomorphism theorem between generalized group G and direct product of two generalized subgroups N and H. These were then compared with the existing results in classical groups to bring out similarities and differences between generalized groups and classical groups. It was discovered that the idea of generalized group gives similar results as in classical group. If G is a generalized group and H is a generalized subgroup of G, it was shown that a Bol groupoid A which is a Cartesian product of H x G defined by (h1, g1)• (h2, 92) = (h1h2, h2g1h2-1g2), V (h1, 91), (h2, 92) E A can be constructed to be non associative and to satisfy Bol identity (xy . z)y = x(yz . y) provided H is an Abelian generalized subgroup of G. Furthermore, it was established that if a non Abelian generalized group G obeys cancellation laws, then a Bol quasigroup with a left identity element can be constructed.