Mathematics and Statistics Vol. 12(4), pp. 374 - 380
DOI: 10.13189/ms.2024.120409
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Homogeneous Spaces and Induced Transformation Groups of S-Topological Transformation Group

C. Rajapandiyan , V. Visalakshi ^*
Department of Mathematics, College of Engineering and Technology, SRM Institute of Science and Technology, Kattankulathur - 603 203, Tamil Nadu, India

ABSTRACT

This paper explores the homogeneous spaces and induced transformation groups of S-topological transformation group. S-topological transformation group is a structure constructed by concatenating a topological group with a topological space through a semi totally continuous action. It is shown that any map from a topological group to the quotient group of a finite Hausdorff topological group by the isotropy group is surjective, continuous, open and it has been proven that any map from the quotient group of a finite Hausdorff topological group by the isotropy group to the homogenous space is both H-isomorphism and semi totally continuous. Furthermore, an equivariant map has been established between homogeneous spaces and discussed the partial order relation on the family of all Hausdorff homogeneous spaces for a compact Hausdorff topological group. Subsequently, an induced S-topological transformation group is constructed by an induced H-action. For any compact subgroup K of a topological group H, it is verified that any map from the topological spcae Y to the orbit space of K-action is continuous and a K-map. For any H-space, K-map and an induced S-topological transformation group; it is proved that there is a unique semi totally continuous H-map. Additionally, it is shown that for a topological group, a subgroup K of topological group and a K-space, there is a unique H-space and a unique injective K-map and also it is established that for a H-space and a semi totally continuous K-map, there exists a unique semi totally continuous H-map. Finally, it is demonstrated that for a finite Hausdorff topological group, finite Frechet space and a M-space, any map from the orbit space of M-action to is semi totally continuous, for the subgroups M and N of topological group.

KEYWORDS
Topological Transformation Group (TTG), Stopological Transformation Group (S-TTG), Isotropy Group, Homogeneous Spaces, Induced Transformation Groups

Cite This Paper in IEEE or APA Citation Styles
(a). IEEE Format:
[1] C. Rajapandiyan , V. Visalakshi , "Homogeneous Spaces and Induced Transformation Groups of S-Topological Transformation Group," Mathematics and Statistics, Vol. 12, No. 4, pp. 374 - 380, 2024. DOI: 10.13189/ms.2024.120409.

(b). APA Format:
C. Rajapandiyan , V. Visalakshi (2024). Homogeneous Spaces and Induced Transformation Groups of S-Topological Transformation Group. Mathematics and Statistics, 12(4), 374 - 380. DOI: 10.13189/ms.2024.120409.