Quasi-completeness in commutative modular group algebras

Abstract. Let K be a field of characteristic p> 0 and let G be a direct sum of cyclic groups, such that its torsion part is a p-group. If there exists a K-isomorphism KH ∼ = KG for some group H, then it is shown that H ∼ = G. Let G be a direct sum of cyclic groups, a divisible group or a simply presented torsion abelian group. Then KH ∼ = KG as K-algebras for all fields K and some group H if and only if H ∼ = G