This is the problem in the book that I want to prove, but it doesn't seem right.
For example let's say I have a group of $9$ elements. If this group is non-cyclic then every element (except identity) has an order of $3$ (prime) (because of Lagrange's theorem). So there are actually $8$ elements of order $3$.
Theorem seems to work for cyclic groups.
Is there is something I am missing here or this only holds for cyclic groups?