For eact element a Z in the group D4Z compute ordaZ b Is th

For eact \"element\" (?!) a Z in the group D_4/Z, compute ord(aZ). (b) Is the group D_4/Z cyslic? If so, find a generator. If not, explain why not.

Solution

order is four since

D4 is non-abelian, so the quotient D4/Z(D4) cannot be cyclic, hence must be isomorphic to K4. and we can compute that g^2 Z(D4) for all g^2 D4. This shows that every non-identity element of D4/Z(D4) has order two, so D4/Z(D4) ~= K4.

since it is non-abelian it is not cyclic