An isomorphism theorem for commutative modular group algebras

Abstract

For each positive integer n and limit ordinal p , a new class of abelian p-groups, called An(p)-gro\ips, are introduced. These groups are shown to be uniquely determined up to isomorphism by numerical invariants which include, but are not restricted to, their Ulm-Kaplansky invariants. As an application of this uniqueness theorem, we prove an isomorphism result for group algebras: Let H be an An(p)-group and F a field of characteristic p . It is shown that if K is a group such that the group algebras F H and FK are F-isomorphic, then H and K are isomorphic.