A note on a theorem of May concerning commutative group algebras

Abstract:

Let $G$ be a coproduct of $p$-primary abelian groups with each factor of cardinality not exceeding ${\aleph _1}$, and let $F$ be a perfect field of characteristic $p$. If $V(G)$ is the group of normalized units of the group algebra $F(G)$, it is shown that $G$ is a direct factor of $V(G)$ and that the complementary factor is simply presented. This generalizes a theorem of W. May, who proved the result in the case when $G$ itself has cardinality not exceeding ${\aleph _1}$ and length not exceeding ${\omega _1}$.