An isomorphism theorem for commutative modular group algebras

Abstract

For each positive integer n n and limit ordinal μ \mu , a new class of abelian p p -groups, called A n ( μ ) {A_n}(\mu ) -groups, 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 H be an A n ( μ ) {A_n}(\mu ) -group and F F a field of characteristic p p . It is shown that if K K is a group such that the group algebras FH and FK are F F -isomorphic, then H H and K K are isomorphic.