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Let $X$ be a set, and $G$ a group.

Consider a map $g:X\times X \to G$ such that for any $x,y\in X$: $$ g(x,y) = g(y,x)^{-1}, $$

and moreover for any $x,y,z\in X$: $$ g(x,y)g(y,z) = g(x,z). $$

Is it always true that there is a map $h:X\to G$ such that for any $x,y\in X$, $$ g(x,y) = h(x)h(y)^{-1}\quad ? $$

Note: I've just asked a similar question, without the last property.

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    $\begingroup$ choose a $x_0\in X$, set $h(x)=g(x,x_0)$. $\endgroup$ – user8268 May 14 '14 at 20:11
  • $\begingroup$ Wonderful! Thanks. $\endgroup$ – geodude May 14 '14 at 23:09
  • $\begingroup$ Technically I'd called it group-element-valued, not group-valued. $\endgroup$ – blue May 15 '14 at 2:51

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