An isomorphism theorem for commutative modular group algebras

Abstract:

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