Is D4 (the octic group) isomorphic to U(16) (group of units mod 16)?

Question

I know they both have the same cardinality and are not cyclic. As well, I know that both groups have the same number of finite elements.

I know of no way to disprove that they're isomorphic, but am having a hard time finding a certain isomorphism.

Any ideas would be greatly appreciated.

Answer 1

I’m pretty sure that U(16) is Abelian — it inherits commutativity from arithmetic mod 16. So U(16) and D4 are not isomorphic, since D4 is non-Abelian.