RO-NMO 2008 Problem 124

Let $\mathcal G$ be the set of all finite groups with at least two elements.

a) Prove that if $G\in \mathcal G$ , then the number of morphisms $f: G\to G$ is at most $\sqrt [p]{n^n}$ , where $p$ is the largest prime divisor of $n$ , and $n$ is the number of elements in $G$ .

b) Find all the groups in $\mathcal G$ for which the inequality at point a) is an equality.