This concerns one of those "well known" facts, referred to in a recent preprint I've been looking at. In principle it's elementary, but I can't pin down an explicit textbook reference for it. Start with two finite groups $A,B$ and their product $G:=A \times B$, working over a splitting field $K$ for the groups involved with prime characteristic dividing $|G|$. Let $S\_1, \dots, S\_m$ and $T\_1, \dots, T\_n$ be respective sets of representatives of isomorphism classes of simple modules for the group algebras $KA, KB$. In turn let the projective covers (=injective hulls) be respectively $P\_i, Q\_j$. These are the PIMs or
indecomposable projective modules for the two group algebras.


It's a standard observation (found in some books) that there is an obvious isomorphism between $KG$ and the tensor product algebra $KA \otimes\_K KB$, while each group algebra splits into the direct sum (as a left module over itself) of the various PIMs taken with multiplicity equal to the dimension of the corresponding simple module. It's also a standard fact (found in some books) that each $S\_i \otimes T\_j$ is a simple module for $KG$. From these ingredients one can conclude that
$P\_i \otimes Q\_j$ is the corresponding PIM, thereby exhausting all isomorphism classes for $KG$. 



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> Is all of this written down in a self-contained way somewhere?
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Reference for projective covers of direct products of finite groups?
The paper Representations of direct products of finite groups.
Burton Fein
Source: Pacific J. Math. Volume 20, Number 1 (1967), 45-58.


[Link](https://projecteuclid.org/journals/pacific-journal-of-mathematics/volume-20/issue-1/Representations-of-direct-products-of-finite-groups/pjm/1102992967.full)


has what you are looking for and also explains what happens in the case that $K$ is not a splitting field. Look at Theorem 2.2 and the remark following it.


