is this correct

f = S, and # = Ay , Is H a normal subgroup of G. Step1: Understand the Groups * Sy: The symmetric group on 4 elements. It has 41 = 24 elements. Ay: The alternating group on 4 elements (even permutations in $4]. It has = = 12 elements. Step2: 15 / sub of $7 The are twelve claments in a, The group d, is sub roup of by To be a subgroup Ay must satisfy Identity: the identity permutation is (1) in A, Closure: multiplying two even permutation gives an even permutation (so A, is closed) (123] (243] Apply (2 3 4) first, them (1 2 3] 1: (24 3)1 + 1, (123)1 + 2. =1+2 2: (24 3)2 + 4, (123)4 + 4.42+4 3: (243)3 - 2, (123)2+3.3 -3 4: (2 4 3)4- 3, (123)3 - 1.64 - 1 1 - 2 + 4 + 1 This is the cycle [1 2 4) (123)(243) = (124) (124) Inverses: the inverse of an even permutation is aven (so inverse are in A,). (124) = (421) = (214) = (142) step3: Is dy normal in S, [Subgroup TEST] Recall: Ay has 12elements, and S, has 24 elements. The index of Ay in S. = [5:4] - -2 Fact A sub of index 2 is always normal. This is because there are only two left cosets; their union is the whole group, so left and right cosets must be the same. O