MODULAR ABELIAN GROUP ALGEBRAS

Abstract

Suppose FG is the F-group algebra of an arbitrary multiplicative abelian group G with p-component of torsion Gp over a field F of char (F) = p ≠ 0. Our theorems state thus: The factor-group S(FG)/Gp of all normed p-units in FG modulo Gp is always totally projective, provided G is a coproduct of groups whose p-components are of countable length and F is perfect. Moreover, if G is a p-mixed coproduct of groups with torsion parts of countable length and FH ≅ FG as F-algebras, then there is a totally projective p-group T of length ≤ Ω such that H × T ≅ G × T. These are generalizations to results by Hill-Ullery (1997). As a consequence, if G is a p-splitting coproduct of groups each of which has p-component with length < Ω and FH ≅ FG are F-isomorphic, then H is p-splitting. This is an extension of a result of May (1989). Our applications are the following: Let G be p-mixed algebraically compact or p-mixed splitting with torsion-complete Gp or p-mixed of torsion-free rank one with torsion-complete Gp. Then the F-isomorphism FH ≅ FG for any group H implies H ≅ G. Moreover, letting G be a coproduct of torsion-complete p-groups or G be a coproduct of p-local algebraically compact groups, then [Formula: see text]-isomorphism [Formula: see text] for an arbitrary group H over the simple field [Formula: see text] of p-elements yields H ≅ G. These completely settle in a more general form a question raised by May (1979) for p-torsion groups and also strengthen results due to Beers-Richman-Walker (1983).