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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.

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)
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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
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