I read that in order to prove it using groups, we can see that the order of the group of units is equal to Euler's totient function: 
$$|U(n)|= \phi (n)$$
After this, however, the author goes on to say that this implies that $a^{\phi (n)}=1$ for all $a \in U(n)$. The only reason I can make this connection is that $U(n)$ might be cyclic, in which $a$ to the power of the order of the group is the identity. But what if $U(n)$ is not cyclic? How can we make this logical conclusion then?