Exterior product of groups

For groups with compatible actions and with a mapping to a common group used for identification
Suppose $$G$$ and $$H$$ are (possibly equal, possibly distinct) groups with a compatible pair of actions on each other and ...

For groups that are normal subgroups of a big group
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).

Define a compatible pair of actions of $$G$$ and $$H$$ on each other by each acting on the other as conjugation in $$Q$$. The exterior product $$G \wedge H$$ is defined as the quotient group of the tensor product of groups $$G \otimes H$$ for this compatible pair of actions by the normal subgroup generated by all elements of the form $$x \otimes x, x \in G \cap H$$.

The image of the symbol $$g \otimes h$$ in the quotient is denoted $$g \wedge h$$.

In terms of exterior pairing
Whichever of the two definitions above we use, we can define the exterior product of $$G$$ and $$H$$ as the group $$G \wedge H$$ such that for every group $$K$$, each exterior pairing $$G \times H \to K$$ corresponds to a unique group homomorphism $$G \wedge H \to K$$.

Facts

 * Exterior product of finite groups is finite
 * Exterior product of p-groups is p-group