Commutative Group Algebras of Cardinality $${\cal N}$$ 1

Abstract

We let FG be the group algebra of an abelian group G over a field F with characteristic p. Also, we define G p and S(FG) as the groups of all p-primary normed elements in G and FG, respectively. We prove that if G p is Hausdorff and both F and G have cardinalities not exceeding \({\cal N}\)1, then S(FG)/G p is a direct sum of cyclics. Thus G p is a direct factor of S(FG), and in particular G is a direct factor of the group of all normalized units V(FG), provided that the torsion part of G is a p-group. This answers a question posed by us in Hokkaido Math. J. (2000). Moreover we establish that if G is p-splitting, then any F-isomorphism of the group algebras FG and FH implies that H is p-splitting. We also show that if G is of power \({\cal N}\)1 whose p-component G p is a direct sum of torsion-complete groups and F has power p, then the F-isomorphism of FG and FH for any group H yields an isomorphism between G p and H p . In particular, when G is of power \({\cal N}\)1 and is p-mixed of torsion-free rank 1 whose G p is torsion-complete, we have G ≂ H. If F is in power p and G, with cardinality \({\cal N}\)1, is a direct sum of p-local algebraically compact groups such that FG ≂ FH as F-algebras for some group H, then G ≂ H. These statements extend results due to Beers-Richman-Walker (1983), and also partially solve a well-known question raised by May in 1979.