4 How do you verify whether a given subgroup of a group is n

(4) How do you verify whether a given subgroup of a group is normal or not. Let H = {(1), (1, 2, 3), (1, 3, 2)} S4. Show that it is not normal. Find all right cosets of H in S4. Verify that there exists two right cosets whose product is not a right coset.

Solution

H= {(1),(1,2,3),(1,3,2) } is not normal in S4

Take g= (2,4) in S4 and h=(1,2,3) in H

Now ghg^-1 = (2,4)(1,2,3)(2,4) , inverse of (2,4) is (2,4)

(2,4)(1,2,3)(2,4) = (1,4,3) which is not in H

ghg^-1 not belongs to H ,hence H is not normal in G =S4