Let $k$ be a field and $G$ a group. Suppose we have $kG$-modules $A$ and $B$, and we want to consider $A\otimes_k B$ as a $kG$-module via $g(a\otimes b)=ga\otimes gb$. The module axioms are easy enough to verify, but how does one show this action is even well-defined?
Thoughts: Consider the map $\phi_g:A\times B\to A\otimes_k B$ given by $(a,b)\mapsto ga\otimes gb$, where $g$ is just a fixed group element. I want to show this is $kG$-balanced (so that we have a well-defined group homomorphism on $A\otimes_k B$). It's obviously additive in both coordinates, but it's not clear to me why $\phi_g(a,xb)=\phi_g(ax,b)$ for $x\in kG$. This seems to involve passing $g$ by $x$, but that is very suspicious