G. Even/Odd Permutations in Subgroups of SnProve each of the following in Sn: Let a1, ar be distinct even permutations, and beta an odd permutation. Then a1beta, arbeta are r distinct odd permutations. (See Exercise C2.) If beta1, betar are distinct odd permutations, then beta1beta1, beta1 beta2,beta1 beta r are r distinct even permutations. In Sn there are the same number of odd permutations as even permutations. (HINT: Use part 1 to prove that the number of even permutation is the number of odd permutations Use part 2 to prove the reverse of that inequality.) The set of all the even permutation if a subgroup of Sn. (It is denoted by An called the alternating group on n symbols ) Let H be any of Sn H either contains only even permutations, or H contains the same number of odd as even permutations. (Use parts 1 and 2 )