Joint complements and isomorphism of modular Abelian group algebras. (Q2853971)

Let \(F\) be a perfect field of prime characteristic \(p\), let \(G\) be a \(p\)-primary torsion or \(p\)-mixed Abelian group, \(FG\) the group algebra. The following questions are of outstanding interest: (MIP) the modular isomorphism problem, that is, whether \(FG\cong FH\) implies \(G\cong H\); (DFP) the direct factor problem, that is, whether the group of normalized units \(V(FG)=G\times B\); (SDFP) simply presented direct factor problem, that is, whether \(V(FG)=G\times B\) for some simply presented \(p\)-group \(B\). These problems has been investigated recently by the author [in Models, modules and abelian groups. Berlin: Walter de Gruyter. 267-276 (2008; Zbl 1188.16032)].NEWLINENEWLINE The main results of the paper are as follows, constitute crucial steps towards proving (MIP). If (SDFP) holds for a \(p\)-group or for a \(p\)-mixed group \(G\) then (MIP) also holds. (MIP) holds for certain countable \(G\) and separable uncountable \(G\) provided all countable subgroups are coproducts of cyclic groups. A result of independent interest is needed during the proof, namely, a necessary and sufficient condition is given for two direct summands, with one of them having a simply presented complement, of an Abelian group to possess a joint complement. This extends results of \textit{R. Göbel} and the author [from Proc. Am. Math. Soc. 131, No. 9, 2705-2710 (2003; Zbl 1026.20033)].