Power automorphism of elemantary abelian group

Question

I proved that a subgroup A normalizes every subgroup of the minimal normal subgroup $N$, $N$ is an elementary abelian group of order $p^n$, $n>1$. It is clear that A induces a power automorphism group of N. In a number of papers there is a statement that the power automorphism group of $N$ is a cyclic group of order $p−1$. Why this statement is true?

Answer 1

You can think of $N$ as being a vector space of dimension $n$ over the field ${\mathbb F}_p$ of order $p$. So the automorphism group of $N$ is equal to that of $V$, which is the group ${\rm GL}(n,p)$ of $n \times n$ invertible matrices with entries in ${\mathbb F}_p$.

Then a power automorphism $g \mapsto g^t$ of $N$ corresponds to a scalar matrix with $t$ on the diagonal. So the group of power automorphisms is isomorphic to the group of (nonzero) scalar matrices, which is isomorphic to the multip;licative group $({\mathbb F}_p - \{0\},\times)$ of ${\mathbb F}_p$, and this is cyclic of order $p-1$.