Proving that the group $U(30)$ is isomorphic to the group $\Bbb Z_8$.

Question

I am trying to prove that the group $U(30)$ is isomorphic to the group $\Bbb Z_8$.

I know that both groups have an order of $8$ and are both cyclic, but this isn't enough to show that an isomorphism exists between the two.

To show an isomorphism exists between two groups I need to show that there exists a bijection between them and that $F(xy) = F(x)*F(y)$, however, I don't have a physical function to work with.

Does anyone have any ideas?

Answer 1

Since

$$30=2\times 3\times 5$$

is the prime factorization of $30$ and hence $U_{30}$ is isomorphic to $ U_2\times U_3\times U_5$ which is $\mathbb Z_2\times\mathbb Z_4$ and hence the given group is not cyclic.