Crossed pairing

Definition

Suppose $G$ and $H$ are (not necessarily abelian) groups with a compatible pair of actions $\alpha :G\to \operatorname {Aut} (H)$ and $\beta :H\to \operatorname {Aut} (G)$. Suppose $K$ is a group. A crossed pairing from $G,H$ to $K$ is a function $f:G\times H\to K$ satisfying the following:

• $f(g_{1}g_{2},h)=f(g_{1}g_{2}g_{1}^{-1},\alpha (g_{1})h)f(g_{1},h)$ for all $g_{1},g_{2}\in G,h\in H$
• $f(g,h_{1}h_{2})=f(g,h_{1})f(\beta (h_{1})g,h_{1}h_{2}h_{1}^{-1})$ for all $g\in G,h_{1},h_{2}\in H$

If we denote the actions by conjugation of the groups on themselves by $\cdot$, and also denote the actions $\alpha$ and $\beta$ of the groups on each other by $\cdot$, the conditions read as follows:

• $f(g_{1}g_{2},h)=f(g_{1}\cdot g_{2},g_{1}\cdot h)f(g_{1},h)$ for all $g_{1},g_{2}\in G,h\in H$
• $f(g,h_{1}h_{2})=f(g,h_{1})f(h_{1}\cdot g,h_{1}\cdot h_{2})$ for all $g\in G,h_{1},h_{2}\in H$

The notion of crossed pairing is related to the notion of tensor product of groups as follows. There is a natural bijective correspondence:

Crossed pairings $G,H$ to $K$ $\leftrightarrow$ Group homomorphisms $G\otimes H$ to $K$