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Let $G$ be a group and $K:G\times G\rightarrow G$ be defined as $K(g,h)=ghg^{-1}h^{-1}$ and $K'=span(K(G,G))$. Calculate $(K(g,h))^{-1}$. What is the relationship between $K(G,G)$ and $K'$?

It is easy to check that $(K(g,h))^{-1} = K(h,g)$. Thus $K(G,G)$ is closed under inverses. However, I don't quite get what I should deduce about the relationship between $K(G,G)$ and $K'$. Any help or hint as to what I should be looking for is greatly appreciated.

Analysis801
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  • Are they equal? Are they unequal? Is one contained in the other? Are there conditions under which these questions have special answers? Are there counterexamples to these various questions? Things like that. – Lee Mosher Apr 29 '18 at 13:47
  • I could not yet figure out whether $K(G,G)$ is a group or not. So I do not know if they are equal. Clearly, $K(G,G)\subseteq K'$. Also, if $G$ is abelian, both are just ${e}$. But nothing of that uses the inverse calculated before. – Analysis801 Apr 29 '18 at 13:51
  • Examples and counterexamples are key to this investigation. – Lee Mosher Apr 29 '18 at 13:54
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    https://math.stackexchange.com/questions/7811/derived-subgroup-where-not-every-element-is-a-commutator/7885#7885 – ancient mathematician Apr 29 '18 at 14:53

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