UNITS OF COMMUTATIVE MODULAR GROUP-ALGEBRAS

Let F be a perfect field of characteristic p and let G be an arbitrary abelian p-group. The normalized units of the group algebra F(G) are denoted by V(G). The proposition that G is a direct factor of V(G) is a long-standing question, which we call the Direct Factor Problem. It is known that the Direct Factor Problem has an affirmative solution provided that V(G)/G is simply presented. In this paper, we prove that V(G)/G has a nu-basis. It has recently been shown for an abelian p-group of cardinality not exceeding aleph1 that having a nu-basis implies the group is simply presented, but for larger groups this implication is unresolved.