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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) $[A,\langle x\rangle]=\{[a,x]|a\in A\}$.

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    No problem. ${}{}$2012-12-24

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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 $a\in \text{Ker}(a\mapsto [a,x])$?

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    @user53587 Use (a) to prove (b). Hint: What are the relevant groups of a homomorphism? E.g. the kernel is one.2012-12-24