G^2 is a normal subgroup of G, with automorphism

Question

G^2 is generated by g^2, g is in G. My professor says that for all g, and for all automorphisms s of G s(g^2)=(s(g))^2

How can he say that for ALL automorphisms whitout knowing that G^2 is characteristic?

Thanks a lot!

Answer 1

Any automorphism $f$ of $G$ has the following property:

$$f(ab)=f(a)f(b) \quad \forall a,b \in G$$

Thus, we have,

$$f(g^2)=f(gg)=f(g)f(g)=(f(g))^2$$