Exterior product of groups

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).

Define a compatible pair of actions of $G$ and $H$ on each other by each actin 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 $g\wedge h$.