Is $U(2^{n})$ isomorphic to $\mathbb{Z_{2}} \bigoplus \mathbb{Z_{2^{n-2}}}$ if $n\geq3$?
And is $U(p^{n})$ isomorphic to $\mathbb{Z_{p^{n}-p^{n-1}}}$ where $p$ an odd prime?
I'm really wondering about its proof.
It's not my homework. I used this fact when doing my homework. I don't have to prove this fact for my homework. This question is just for curiosity.