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I wish to ask a question. Please forgive me if it is does not make any sense:

@ vuur, @ Jack Schmidt

I was searching on google for "commutator subgroup of semidirect product". Fortunately I found a post commutator subgroup and semidirect product on math stack. There a formula for the commutator subgroup of solvable group $G=N \rtimes H$ in terms of $H$ and $N$. I have never seen such a formula and do not know how to prove it. Can some one help me in proving this formula?

p.s. I was trying commenting on this post but because of math stack policy I could not comment on this post as I require 50 reputation.

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If $G = NH$ with $N \unlhd G$ and $H \le G$, then you can use the commutator laws to show that the derived subgroup $G' = [G,G] = [N,N][N,H][H,H]$ and so if $G = N \rtimes H$ then $G' = ([N,N][N,H]) \rtimes H'$. This shows also that $N \cap G' = [N,N][N,H]$.

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