Sn is all permutations on n elementsg which is a group under

S_n is all permutations on n elements_g which is a group under composition. An is the set of all even permutations on n elements. So A_n S_n A_n is a subgroup of S_n by showing the identify exists, the group is closed, inverses exist.

Solution

Let A be a nonempty set, and let S_nbe the collection of all permutations of A.

Then S_n is a group under the composition

Proof: Clearly, o defines a binary operation on S.

We need to check the three axioms G1, G2, G3 of the group.

(Associative property G1) We know composition is associative.

(Identity property G2) Consider the mapping : A A such that for all x A, (x) = x.

Then for any S one has o = o = . Thus, is the identity element on S.

( Inverse property G3) For any permutation S, its inverse 1 satisfies ¹ o = o ¹ = . Thus, ¹ is the inverse of with respect to the group operation.

S_nis satisfy all property of group. S_n is group.