A Study On Algebraic Properties of a Finite Permutation Groups

This paper presents a systematic study of the algebraic properties of permutation groups, with particular emphasis on the symmetric group S_n. We begin by recalling the definition of a permutation of a finite set and introduce the standard two-row and cycle notations. The structure of the symmetric group is examined, including its order, the identity and inverse elements, and the non-commutativity of S_3. Cycle decompositions, transpositions, and the composition of cycles are treated in detail with explicit computations. The work consolidates fundamental results that form the foundation for further investigations into the alternating group and applications in algebra, combinatorics, and related fields.