Tensor Factorizations of Group Algebras and Modules Original Research Article

Here a group algebra is always the group algebra of a finite group over a commutative field. We consider connections between three kinds of factorizations: writing the group as a direct product of subgroups; writing the group algebra as a tenser product of subalgebras; and writing the regular module (the group algebra viewed as a module over itself) as a tenser product of modules. In the principal result the field has prime characteristic, the group order is a power of this prime, and the group is abelian. If in these circumstances the regular module is isomorphic to a tenser product of two modules, then the group has a direct decomposition with one (direct factor) subgroup acting regularly on one of the (tenser factor) modules and the other subgroup acting regularly on the other module. Moreover, the module varieties of the tenser factors must be linear subspaces of the vector space which is the variety of the trivial module, and the two subspaces must form a direct decomposition of that space.