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Let the homomorphism $f:H \rtimes K \rightarrow K$ be defined by $f(hk)=k$.

Now, I will construct the homomorphism $f: [H \rtimes K ,H \rtimes K ] \rightarrow [K,K]$. How to find the kernel of $f$?. Is the kernel isomorphic with $[H,H]$?

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    The kernel certainly contains $[H,H]$; but it is in general larger. For example, if $h\in H$ and $k\in K$ do not commute, then $[h,k]$ maps to $[1,k]=1$, but $[h,k]\notin[H,H]$. More generally, a generator of $[H\rtimes K,H\rtimes K]$ of the form $[hk,h'k']$ will lie in the kernel if and only if $[k,k'] = 1$.2012-04-13

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