Boolean powers of simple groups

We prove that factor groups of cartesian powers of finite non-abelian simple groups cannot be countably infinite. This is not our main result, but it had been our original aim. The proof is based on a similar fact concerning er-complete Boolean algebras, and on a representation of certain subcartesian powers of a group in its group ring over a Boolean ring. This representation, to which we give the name "Boolean power", will be our central theme, and we begin with it.