Soft prime groups, strong conjugation and applications of Sylow Theorems to soft groups

Abstract

In classical algebra, p-groups, conjugate groups and Sylow Theorems are of great importance to understand the arbitrary finite group structures. Our interest, in this paper, is to transfer these important structures to soft group theory. First, we define soft prime group, conjugate group and soft conjugate group. Then, we examine their properties under group mappings, group homomorphisms and soft homomorphisms. Also, as a strong case of conjugate group, strong conjugate group is defined and the relationship between the conjugation and strong conjugation is derived and it is showed that strong conjugation is an equivalence relation on the set of all soft groups over G with the parameter set A. Additionally, we convey Cauchy's Theorem to soft groups. Moreover, in order to understand the structure of an arbitrary finite soft group, we define soft Sylow p-subgroup and obtain the corresponding Sylow Theorems in soft group theory with this concept. By this way, we bring a new aspect to soft group theory by expanding the theory with the fundamental concepts.