A New Type of Soft Group: Soft Symmetric Difference Group with Group Theory Applications

In this paper, a new type of soft group called the soft symmetric difference group (SSD-group) is introduced and systematically developed. This structure is constructed by integrating soft set theory with group theory through the symmetric difference operation and set inclusion. Fundamental concepts such as characteristic soft symmetric difference groups, soft symmetric difference subgroups, normal soft symmetric difference subgroups, soft normalizers, and soft cosets are defined, and their essential algebraic properties are investigated. Several characterizations of soft normality are also established through these concepts. Various axiomatic results are obtained, providing necessary and sufficient conditions for a soft set to form an SSD-group. Furthermore, soft quotient (factor) groups of SSD-groups are introduced and their structural properties are examined in detail. The relationship between SSD-group theory and classical group theory is also established through several corresponding concepts. Illustrative examples are provided to demonstrate the applicability and internal consistency of the proposed framework. Overall, the results obtained in this study extend existing soft group structures and contribute to the development of algebraic theory within the context of soft sets, while also providing a foundation for further generalizations to other algebraic frameworks such as semigroups, rings, and fields.