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In general, the group decompositions are formulated by employing automorphisms and semidirect products to determine continuity and compactification properties. This paper proposes a set of constructions of novel topological decompositions of groups and analyzes the behaviour of group actions under the topological decompositions. The proposed topological decompositions arise in two varieties, such as decomposition based on topological fibers without projections and decomposition in the presence of translated projections in topological spaces. The first variety of decomposition introduces the concepts of topological fibers, locality of group operation and the partitioned local homeomorphism resulting in formulation of transitions and symmetric surjection within the topologically decomposed groups. The reformation of kernel under decomposed homeomorphism and the stability of group action with the existence of a fixed point are analyzed. The first variety of decomposition does not require commutativity maintaining generality. The second variety of projective topological decomposition is formulated considering commutative as well as noncommutative projections in spaces. The effects of finite translations of topologically decomposed groups under projections are analyzed. Moreover, the embedding of a decomposed group in normal topological spaces is formulated in this paper. It is shown that Schoenflies homeomorphic embeddings preserve group homeomorphism in the decomposed embeddings within normal topological spaces. This paper illustrates that decomposed group embedding in normal topological spaces is separable. The applications aspects as well as parametric comparison of group decompositions based on topology, direct product and semidirect product are included in the paper.

Projective and Non-Projective Varieties of Topological Decomposition of Groups with Embeddings