A Study on Sylow Theorems for Finding out Possible Subgroups of a Group in Different Types of Order

This paper aims at treating a study on Sylow theorem of different algebraic structures as groups, order group, subgroups, along with the associated notions automorphisms group dihedral split extensions groups and vector spaces arises from varying properties real complex numbers. We must have used theorems this work when it's generalized. Here we discuss possible subgroups in types which will give us practical knowledge to see applications theorems. In structures, deal operations addition multiplication those greater than, less than so on. It is through that realize importance some definitions like exact sequences p-subgroup semi-direct product. Thus it has been found necessary convenient these detail. situations, where was given situation satisfies basic axioms structure having already known structure. Finally, find out for abelian non-abelian cases.