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On the Kurosh theorem. (English. Russian original) Zbl 0917.20023

Algebra Logika 37, No. 4, 381-393 (1998); translation in Algebra Logic 37, No. 4, 215-222 (1998).
The Kurosh theorem on subgroups of free products is an important structural result, repeatedly reproven by different methods. In all proofs of the theorem, use was made of a combinatorial description of free products rather than of their categorical properties. In the category of profinite groups, an acceptable analog of the Kurosh theorem was obtained by D. Haran [Proc. Lond. Math. Soc., III. Ser. 55, 266-298 (1987; Zbl 0666.20015)], who used the notion of wreath product and categorical properties of the groups. The author of the article under review deals with abstract (not profinite) groups and with their homomorphisms. The notion of a projective family of subgroups is introduced which behaves well under passage to subgroups and, thus, relates to the notion of a free product. This provides a new proof of the Kurosh theorem on subgroups of a free product in which the use is actually made of purely categorical properties of free products. The author also restates some of the results for profinite groups.

MSC:

20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
20E07 Subgroup theorems; subgroup growth
20A15 Applications of logic to group theory
20J15 Category of groups
18B40 Groupoids, semigroupoids, semigroups, groups (viewed as categories)
20E18 Limits, profinite groups

Citations:

Zbl 0666.20015
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