Computing the subgroups of $\mathbb{Z}_n^*$.

Question

I'm wondering if there is a way to list all the subgroups of $\mathbb{Z}_n^*$, just as there is with $\mathbb{Z}_n^+$ and $D_n$, yet I can't seem to find anything online.

Answer 1

The group is cyclic if and only if $n=1,2,4,p^k$ or $2p^k$ for $p$ an odd prime.

In the other cases, you still have a finite abelian group of order $\phi(n)$, so you can employ the structure theorem.

The group of units functor respects products, so this is feasible.

For instance, $\Bbb Z_{21}^×\cong\Bbb Z_3^××\Bbb Z_7^×\cong\Bbb Z_2×\Bbb Z_6$. So the subgroups are easy to find.