Tensor product of p-groups is p-group

Statement

Finite p-groups version

Suppose $G$ and $H$ are finite p-groups for some prime number $p$ with a compatible pair of actions on each other. Then, the tensor product of groups $G\otimes H$ with respect to the pair of actions is also a finite p-group for the same prime $p$.

Arbitrary p-groups version

Suppose $G$ and $H$ are p-groups for some prime number $p$ with a compatible pair of actions on each other. Then, the tensor product of groups $G\otimes H$ with respect to the pair of actions is also a p-group for the same prime $p$.

Related facts

• Tensor product of finite groups is finite

References

Journal 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}