Exterior product of finite groups is finite

Statement

Suppose $G,H$ are both finite groups that arise as normal subgroups inside a group $Q$. (We can assume without loss of generality that $Q$ is finite, because we can replace $Q$ by the subgroup $GH$ if necessary, whose size is given by the product formula).

Then, the exterior product of groups $G\wedge H$ is a finite group.

Related facts

• Exterior square of finite group is finite
• Exterior product of p-groups is p-group
• Tensor product of finite groups is finite
• Tensor product of p-groups is p-group

References

Textbook references

• The non-abelian tensor product of finite groups is finite by Graham Ellis, Journal of Algebra, ISSN 00218693, Volume 111, Page 203 - 205(Year 1987): ^{Official copy (Elsevier ScienceDirect, via DOI)More info}