Abstract

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.

Highlights

  • The applications of group theory are found in the identification of symmetries and in designing cryptographic protocols [1,2]

  • The applications aspects as well as parametric comparison of group decompositions based on topology, direct product and semidirect product are included in the paper

  • As a consequence, every closed subgroup of a group becomes separable if the group is a compact separable topological group

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Summary

Introduction

The applications of group theory are found in the identification of symmetries and in designing cryptographic protocols [1,2]. Topology and group theory exhibit interplay if the underlying space is considered to be a topological space. Topological groups are special groups with topological properties in underlying spaces. The separable topological spaces are considered while analyzing topological groups, as well as their subgroups, with closure properties [4]. The constructions of various types of projection in topological spaces can be formulated with algebraic and geometric. The geometric variety of projective spaces with distribution of singular points is proposed in [5]. The factoriality of geometric hypersurface under projection is constructed up to d-degree containing singular points. The Galois projective spaces (finite variety) have applications in coding theory, group theory and cryptology [6]. The motivation for the formulation of topological decomposition of groups is described

Compactification and Projections
Decomposition and Soft Sets
Motivation
Preliminary Concepts
Decomposition Varieties
Non-Projective Decomposition
Topological Space Partition
Group Homeomorphisms
Schoenflies Homeomorphic Embeddings
3.1.10. Decomposed Group Embedding
Topological Decomposition with Projection
Projection of Decomposition
Noncommutative Projection
Finiteness of Translated Projection
Inflationary Bounded Translation
Analytical Properties of Decomposition Varieties
Properties of Non-Projective Variety
Properties of Projective Variety of Decomposition
Application Aspects and Comparison
Conclusions
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