Commutative group algebras of cardinality \(\aleph_1\) (Q1849295)

Let \(FG\) be the group algebra of an Abelian group over a perfect field and \(V(FG)\) the group of normalized units of \(FG\). It is a long-standing conjecture going back to May that \(V(FG)/G\) is simply presented provided that \(G\) is a \(p\)-group. \textit{W. May} [Contemp. Math. 93, 303-308 (1989; Zbl 0676.16010)], \textit{P. Hill} and \textit{W. Ullery} [Proc. Am. Math. Soc. 110, No. 1, 59-63 (1990; Zbl 0704.20007)] have independently proved this conjecture if \(G\) is an Abelian \(p\)-group of length \(\leq\omega_1\) and \(|G|\leq\aleph\). The author extends this result to an arbitrary Abelian group \(G\) provided that \(|G|\leq\aleph\) and the \(p\)-Sylow subgroup \(G_p\) of \(G\) is separable: if \(S(FG)\) is the \(p\)-Sylow subgroup of \(V(FG)\), then \(S(FG)/G_p\) is a direct sum of cyclic groups. The author applies this result to the isomorphism problem for the group algebras of groups with the above properties.