Let $A$ be an abelian normal subgroup of $G$ and $x\in G$. How can we prove the following?
(a) The mapping $A\mapsto A$ given by $a \mapsto [a,x]$ is a homomorphism.
(b) $[A,\langle x\rangle]=\{[a,x]|a\in A\}$.
Let $A$ be an abelian normal subgroup of $G$ and $x\in G$. How can we prove the following?
(a) The mapping $A\mapsto A$ given by $a \mapsto [a,x]$ is a homomorphism.
(b) $[A,\langle x\rangle]=\{[a,x]|a\in A\}$.
Hint: $[ab,x]=[a,x]^b[b,x]$
Stronger Hint: $[a,x^2]\in A$ because $A$ is normal. What can you say about $[a,x^2]$ when $a\in \text{Ker}(a\mapsto [a,x])$?