Isomorphism of commutative group algebras of closed \(p\)-groups and \(p\)-local algebraically compact groups (Q2782619)

\textit{W. May} [in Proc. Am. Math. Soc. 76, 31-34 (1979; Zbl 0388.20041)] asked whether for a \(p\)-torsion Abelian group being closed is an invariant property for the group algebra over an arbitrary field of characteristic \(p\). Partial answers to this problem have been given by the author [in Rend. Sem. Mat. Univ. Padova 101, 51-58 (1999; Zbl 0959.20003); Southeast Asian Bull. Math. 25, No. 4, 589-598 (2002)]. This paper completely settles the question and provides new perspectives for investigating algebraically compact groups developing a new technique.NEWLINENEWLINENEWLINEThe central theorem of the paper is as follows. Let \(G\) be a closed Abelian \(p\)-group or a \(p\)-local algebraically compact Abelian group. Suppose that the so called modified direct-factor problem (due to \textit{W. May} [Contemp. Math. 93, 303-308 (1989; Zbl 0676.16010)]) is fulfilled, that is, \(K\) is a perfect field of characteristic \(p\) and \(S(KG)/G_p\) is totally projective, where \(G_p\) and \(S(KG)\) are the Sylow \(p\)-subgroup of \(G\) and the normalized group of units of \(KG\), respectively. Then \(KH\cong KG\) as \(K\)-algebras if and only if \(H\cong G\).