Tensor product of p-groups is p-group

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