Given the multiplicative group G=U(32)={1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31} and a normal subgroup H={1,15} of G prove whether the quotient group G/H is isomorphic to $Z_{8}$ or to $Z_{4}$ $ \oplus$ $Z_{2}$.
The quotient group is G/H = {H,3H,5H,7H,17H,19H,21H,23H} with identity H.The order of the elements of G/H is respectively (1,8,8,4,2,8,8,4). $Z_{8}$ = {0,1,2,3,4,5,6,7} has four elements with order 8, one element with order 2 and two elements with order 4.
In my book the answer to this question is that G/H is isomorphic to $Z_{4}$ $ \oplus$ $Z_{2}$.
I do not understand why? Can you help me...
Hint:
Can you find any element of $G$ whose fourth power is not in the subgroup?
If so, this would suggest that there are cosets with an order greater than four in the quotient group.
If not, every coset has order at most $4$ in the quotient group.