Modular group algebras of coproducts of countable Abelian groups
Let \(G\) be a coproduct of countable groups, let \(G_p\) be the \(p\)-component of \(G\) and let \(F\) be a perfect field of characteristic \(p\). Denote by \(FG\) the group algebra of \(G\) over \(F\) and by \(S(FG)\) the \(p\)-component of the group \(V(FG)\) of normalized units in \(FG\). In the paper it is proved that \(G_p\) is a direct factor of \(S(FG)\) and \(S(FG)/G_p\) is a coproduct of countable groups. Besides, the \(F\)-isomorphism of \(FH\) and \(FG\) for any group \(H\) implies the following: (1) \(H_p\) is isomorphic to \(G_p\), (2) if the torsion subgroup \(tG\) of \(G\) is a \(p\)-group then \(G\) is a direct factor of \(V(FG)\), \(tH\) and \(tG\) are isomorphic and there exists a totally projective group \(T\) with length not exceeding the first uncountable ordinal such that \(H\times T\) and \(G\times T\) are isomorphic. These results are analogous to results obtained by \textit{P. Hill} and \textit{W. Ullery} [5, Commun. Algebra 25, No. 12, 4029-4038 (1997; Zbl 0901.16012)]. Some more remarks of the reviewer. The author does not elucidate two main facts in the proof of his two theorems. Namely, we read: ``... we must prove only that \(N_\alpha G_p\) is nice (or can be expanded in a nice subgroup) in \(S(FG)\). But the latter follows automatically from the technique described in [5, Theorems 4.2 and 5.6]'', p. 259, lines \(14^-\)-\(11^-\) and in the proof of the isomorphism theorem ``... hence the same holds for \(H_p\) by application of a slight modification of [5, Theorem 5.6]'', p.260, lines \(12^-\)-\(11^-\). The two facts indicated are essential, they do not follow automatically or by slight modification and should be clarified in the paper. On the other hand, the expression in the brackets of the first citation cannot be used in the proof and it is entirely redundant because every subgroup can be expanded in a nice subgroup. It is not noted, that the proof of Lemma 4 is analogous to a similar lemma of \textit{N. A. Nachev} [Lemma 4, Houston J. Math. 22, No. 2, 225-232 (1996; Zbl 0859.16025)].
scientific article; zbMATH DE number 1532959