normal subgroup, diagonal embedding.

Question

Suppose that $A$ is a normal subgroup of $G$, and $A$ is abelian. Let $i:A\rightarrow A\times G$ a morphism of groups given by $a\mapsto (a,a)$. Is is clear that $i(A)$ is a normal subgroup of $A\times G$ ?

Answer 1

It's not true. Take e.g. $A = C_3$, the cyclic group of order three, viewed as the set of rotations inside $G = D_6$, the symmetry group of the triangle.

It is, however, true that $i(A)$ is normal if and only if $i(A)$ is central, if and only if $A$ is central.