Everything related to the computational difficulty of factorizing integers should give you an answer. For instance, it is computationally easy to compute [Euler's totient function](https://en.wikipedia.org/wiki/Euler%27s_totient_function) for powers of primes, but it is computationally difficult to compute it for arbitrary integers. As a consequence, it is trivial to produce a generator of a multiplicative group of prime order (any nontrivial element would do), but it is computationally very difficult to produce a generator of the multiplicative group of $\mathbb Z / n \mathbb Z$ for composite $n$ (because its order is $\varphi (n)$).