Frattini subgroup of $p$-groups and the automorphisms

Question

About $p$-groups, I saw on Wikipedia that:

Every automorphism of $G$ induces an automorphism on $G/Φ(G)$, where $Φ(G)$ is the Frattini subgroup of $G$. The quotient $G/Φ(G)$ is an elementary abelian group and its automorphism group is a general linear group, so very well understood.

I just don’t get it. Could you explain it?

1. How does a automorphism of $G$ induces an automorphism on $G/Φ(G)$?
2. Why the quotient $G/Φ(G)$ is an elementary abelian group?
3. Why the automorphism group of $G/\Phi(G)$ is a general linear group?

Answer 1

1. $\Phi(G)$ is a characteristic subgroup. That means that for every automorphism $f$ of $G$, $f(\Phi(G))=\Phi(G)$. If $N$ is a normal subgroup of $G$, and $f$ is an an automorphism such that $f(N)=N$, then $f(gN) = f(g)N$ gives an automorphism of $G/N$.

2. Because every maximal subgroup of a $p$-group contains $G’$ and $G^p$, so $\Phi(G)$ contains $G’G^p$. The quotient is therefore abelian and of exponent $p$.

3. No, it does not say that “the automorphism group of a $p$-group is a general linear group.” Read again what it says. It says the automorphism group of $G/\Phi(G)$ is a general linear group. Hint: an elementary abelian $p$-group is a vector space over sometbhing.