Find a proper subgroup H of G S4 and a proper subgroup K of

Find a proper subgroup H of G = S_4 and a proper subgroup K of H such that K is a normal subgroup of H and H is a normal subgroup of S_4, but K is not a normal subgroup of S_4 that is too difficult, you can get 10 points by solving the same problem with G = A_4 instead. Be sure to prove that your groups have the desired properties, full credit only if all needed proofs are given.

Solution

solution:

given that G =S₄.is a symmetric group and H is a subgroup of G .so we have H= A₄ is a subgroup of S₄.

now K is a norma subgroup of H .so we have cylic subgroup of A₄ is ( e,(1 2),(3 4)) which is normal subgroup therefore

K= ( e,(1 2),(3 4) ) implise that K is normal subgroup of H .but K is not normal subgroup of S₄.