Question 30
Let $p$ be a prime number. Consider a commutative finite group $G$, with at least three elements, identity element $e$, and so that $x^p = e$, for every $x \in G$. Denote $Aut(G)$ the group of the automorphisms of $G$. Prove that:
a) $|G| = p^n$, where $n$ is the smallest cardinality of a set of generators of $G$;
b) $|Aut(G)| = (p^n - 1)(p^n - p) \dots (p^n - p^{n-1})$.
a) $|G| = p^n$, where $n$ is the smallest cardinality of a set of generators of $G$;
b) $|Aut(G)| = (p^n - 1)(p^n - p) \dots (p^n - p^{n-1})$.