Indian Journal of Science and Technology

Abstract

Objectives: The primary goal of this study is to present the concept of a strong G 􀀀 group as a generalization of G 􀀀 group. Methods and Findings: We have investigated some of the properties of the G􀀀 group and extended it to introduce the idea of a strong G􀀀 group. Novelty: Every strong G􀀀 group is a G 􀀀 group, but not all G􀀀 groups are strong G􀀀 groups. Further if G is a non-empty G􀀀 semigroup and for all a;b e G, the equations aax = b and yaa = b for all x; y e G and for all a e G have unique solutions in G, then G is a strong G􀀀 group. Also, we characterize that non-empty subsetH of a strong G􀀀 groupG is a strong G􀀀 subgroup if and only if for all a;b eH;aac e H for all a e G where c is strong inverse of b in G. Finally, we prove that the intersection of two strong G􀀀 subgroups is again a strong G􀀀 subgroup and the center of strong G􀀀 group C(G) is also a G􀀀 subgroup.

Keywords: Semigroup; Strong G 􀀀group; Strong G 􀀀subgroup; Centre of G 􀀀group