Ershov, Yu. L. Projective products of profinite groups. (English. Russian original) Zbl 0820.20032 Algebra Logic 30, No. 6, 417-426 (1991); translation from Algebra Logika 30, No. 6, 638-651 (1991). The definition of a projective product of profinite groups is given in the paper. Then the following nice characterization of projective profinite products is proved: let \(G\) be a free profinite product of profinite groups \(G_ t\) on a profinite space \(T\) of indices; then a group \(H\) is a projective product of profinite groups \(G_ t\) if and only if \(H = \langle G_ t^{\lambda(t)} \mid t\in T\rangle\) is a closed subgroup of \(G\) generated by conjugates \(G_ t^{\lambda(t)}\) for some continuous map \(\lambda : T \to G\).It follows from the characterization that projective products of the profinite groups \(G_ t\) can be ordered naturally as subgroups of the free profinite product \(G\). In this sense the free product \(G\) of the profinite groups \(G_ t\) is the largest projective product. The author proves also the existence of a smallest projective product under the condition that all \(G_ t\) are nontrivial. Reviewer: P.Zalesskij (Minsk) MSC: 20E18 Limits, profinite groups 20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations 18G05 Projectives and injectives (category-theoretic aspects) Keywords:free products of profinite groups; projective product of profinite groups; projective profinite products; free profinite products; free products PDFBibTeX XMLCite \textit{Yu. L. Ershov}, Algebra Logic 30, No. 6, 417--426 (1991; Zbl 0820.20032); translation from Algebra Logika 30, No. 6, 638--651 (1991) Full Text: DOI References: [1] O. V. Mel’nikov, ”Subgroups and homologies of free products of profinite groups,” Izv. Akad. Nauk SSSR, Ser. Mat.,53, No. 1, 97–120 (1989). · Zbl 0671.20025 [2] D. Haran, ”On closed subgroups of free products of profinite groups,” Proc. London Math. Soc.,55, 266–298 (1987). · Zbl 0666.20015 [3] E. Binz, J. Neikirch, and G. H. Wenzel, ”A subgroup theorem for free products of profinite groups,” J. Algebra,19, 104–109 (1971). · Zbl 0232.20052 · doi:10.1016/0021-8693(71)90118-9 [4] Yu. L. Ershov, ”Invariant generability,” Sib. Mat. Zh.,29, No. 5, 109–111 (1988). · Zbl 0656.53007 · doi:10.1007/BF00975022 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.