نظریه ساختاری فراگروه

Subject Areas : Algebra

Monireh Aliabadi ¹ , Gh. Moghaddasi ² , Parvaneh Zolfaghari ³

1 - Hakim Sabzevari university
2 - Department of Natural Resources, Savadkooh Branch, Islamic Azad University, Mazandaran, Iran
3 - دانشگاه فرهنگیان, گروه ریاضی

Received: 2023-05-24 Accepted : 2023-12-19 Published : 2025-04-08

Keywords: cyclic ultra-groups, solvable ultra-groups, normal ultra-group, category of ultra-group, zassenhaus lemma,

Abstract :

In this article, the study of ultra-groups, whose structure is based on the transversal properties of subgroup of a group is investigated. Firstly, introductory concepts are explained. Subsequently, the importance of the category, some of its properties, such as completeness, and cocompleteness, retraction and coretraction of morphism are examined and shown that in this category, every epimorphism is regular. Additionally, a functor between two category of groups and category of ultra-groups is defined, and the structure of properties like iso-dense, equivalence, and isomorphism for this functor are analyzed. Furthermore, the concept of generated and cyclic ultra-groups is intrduced. Given the significance of cyclic ultra-groups, it is proven that cyclic ultra-groups are not a variety and the validity of the Cauchy and Sylow theorems is examined. Moreover, the normalizer as well as conjugate of an element in ultra-groups are introduced, and the solvability conditions of the ultra-groups are investigated. Finally, the theorems of Jordan-HŐlder, Schreier and Zassenhaus lemma are proved for ultra-groups.

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