Isomorphism of commutative group algebras of closed $p$-groups and $p$-local algebraically compact groups

Abstract

Let G be an abelian group and let K be a field of charK = p > 0. It is shown via a universal algorithm that if the modified Direct-Factor Problem holds, then the K-isomorphism KH ≅ KG for some group H yields H ≅ G provided G is a closed p-group or a p-local algebraically compact group. In particular, this is the case when G is closed p-primary of arbitrary power, or G is p-local algebraically compact with cardinality at most N 1 and K is in cardinality not exceeding N 1 . The last claim completely settles a question raised by W. May in Proc. Amer. Math. Soc. (1979) and partially extends our results published in Rend. Sem. Mat. Univ. Padova (1999) and Southeast Asian Bull. Math. (2001).