Question: Group theory is a wide, autonomous area of mathematics which affects almost any other area of pure mathematics. For computer science is of importance too. It is fundamental for areas such as Computational Complexity, Algorithm, Cryptography, & Combonatorics to name a few. The rest of this question focuses on some particular groups, known as permutation groups.

Fix a set Ω = {1, 2, 3,....., n}. A permutation π on Ω is a bijection (1-1 & onto), π : Ω® Ω. Observe that the composition of two permutations is also a permutation. Here is a convenient way to denote permutations. Example {in this notation same column element from the first row are mapped to the corresponding element of the second row}:

Answer the following:
Fix n € N. Show that the set of sn of all permutations over [1, 2,........n] forms a group under the operation of composition. That is, show that composition is a binary operation over the set of permutations, & the three axioms - mentioned above - are satisfied.