Existence of a permutation consisting of exactly two cycles for odd prime

Question

Is it true that if p is an odd prime number, then there exists some integer a such that the permutation of Ф(p) induced by multiplication by a mod p consists of exactly 2 cycles?

Answer 1

It is a basic fact of group theory that for every odd prime number $p$ the group $\Phi(p)=(\Bbb{Z}/p\Bbb{Z})^{\times}$ is cyclic of order $p-1$. For a proof, see this question. So if $g$ is a generator, then multipication by $g$ is represented by a $p-1$-cycle, and hence multiplication by $g^2$ is represented by its square, the product of two disjoint $\frac{p-1}{2}$-cycles.