Isomorphism of modular group algebras of direct sums of torsion-complete Abelian \(p\)-groups
The author proposes the following result (Theorem 3.1). Let \(F\) be a prime field of positive characteristic \(p\) and let \(G\) be a direct sum of torsion-complete Abelian \(p\)-groups such that each direct factor has cardinality not exceeding the first uncountable cardinal. Then the isomorphism of the group algebras \(FG\) and \(FH\) as \(F\)-algebras for some group \(H\) implies an isomorphism of \(H\) and \(G\). The necessity of Proposition 2.4 is proved falsely (p. 54, lines 8 and 9). This inaccuracy cannot be corrected if the same idea of proof is utilized. The group \(G\) of Proposition 2.5 must be torsion because otherwise its proof in the cases 2) and 3) is not true. The author groundlessly assumes in the proof of Proposition 2.6 that \(G^{(\alpha)}=G_\alpha\) (p. 56, line 1). Finally, the necessity of Proposition 2.4 is used in the proof of Proposition 2.6 for many factors and the last proposition is exploited in the proof of Theorem 3.1. Therefore, Theorem 3.1 is not proved.
scientific article; zbMATH DE number 1319592
scientific article