# Group-valued “skew-symmetric” map with composition property.

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.

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