Is multiplying every element in a group by a fixed element a group automorphism?

Question

Is multiplying every element in a group by a fixed element a group automorphism?

By this I mean taking every $g \in G$ for some group $G$ and mapping it to $ag$ for some fixed $a$ (that may or may not be in $G$).

Answer 1

Yes, if $g$ is the identity element of $G$ and no otherwise. In fact, $g\neq e_G\implies g.e_G\neq e_G$, whereas every group homomorphism maps the identity element into the identity element.