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:


 * $$f(g_1g_2,h) = f(g_1g_2g_1^{-1},\alpha(g_1)h)f(g_1,h)$$ for all $$g_1,g_2 \in G, h \in H$$
 * $$f(g,h_1h_2) = f(g,h_1)f(\beta(h_1)g,h_1h_2h_1^{-1})$$ 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:


 * $$f(g_1g_2,h) = f(g_1 \cdot g_2,g_1 \cdot h)f(g_1,h)$$ for all $$g_1,g_2 \in G, h \in H$$
 * $$f(g,h_1h_2) = f(g,h_1)f(h_1 \cdot g, h_1 \cdot h_2)$$ 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 $$K$$