TENSOR FACTORIZATIONS OF GROUP-ALGEBRAS AND MODULES

Here a group algebra is always the group algebra of a finite group ove r a commutative field. We consider connections between three kinds of factorizations: writing the group as a direct product of subgroups; wr iting the group algebra as a tenser product of subalgebras; and writin g the regular module (the group algebra viewed as a module over itself ) as a tenser product of modules. In the principal result the field ha s 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 di rect decomposition with one (direct factor) subgroup acting regularly on one of the (tenser factor) modules and the other subgroup acting re gularly on the other module. Moreover, the module varieties of the ten ser factors must be linear subspaces of the vector space which is the variety of the trivial module, and the two subspaces must form a direc t decomposition of that space. (C) 1995 Academic Press, Inc.