Exterior pairing

Definition
Suppose $$G,H$$ are (possibly equal, possibly distinct) normal subgroups of some group $$Q$$. (Note that it in fact suffices to assume that they normalize each other, but there is no loss of generality in assuming they are both normal, because we can replace the common parent group by a smaller one in which they are both normal).

Suppose $$K$$ is a group.

Define a compatible pair of actions of $$G$$ and $$H$$ on each other by each acting on the other as conjugation in $$Q$$.

An exterior pairing is a function $$f:G \times H \to K$$ that satisfies both these conditions:


 * It is a crossed pairing for this compatible pair of actions.
 * $$f(x,x)$$ is the identity element for all $$x \in G \cap H$$