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

Soft prime groups and applications of Sylow Theorems to soft groups

Résumé

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.