Abstract
The Magnus embedding is well known: given a group A=F/R, where F is a free group, the group F/[R, R] can be represented as a subgroup of a semidirect product AT, where T is an additive group of a free Z A-module. Shmel’kin genralized this construction and found an embedding for F/V(R), where V(R) is the verbal subgroup of R corresponding to a variety V. Later, he treated F as a free product of arbitrary groups, and on condition that R is contained in a Cartesian subgroup of the product, pointed out an embedding for F/V(R). Here, we combine both these Shmel’kin embeddings and weaken the condition on R, by assuming that F is a free product of groups Ai (iεI) and a free group X, and that its normal subgroup R has trivial intersection with each factor Ai. Subject to these conditions, an embedding for F/V(R) is found; we cell it the generalized Shmel’kin embedding. For the case where V is an Abelian variety of groups, a criterion is specified determining whether elements of AT belong to an embedded group F/V(R). Similar results are proved also for profinite groups.
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Supported by RFFR grant No. 99-01-00567.
Translated fromAlgebra i Logika, Vol. 38, No. 5, pp. 598–612, September–October, 1999.
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Romanovskii, N.S. Shmel’kin embeddings for abstract and profinite groups. Algebr Logic 38, 326–334 (1999). https://doi.org/10.1007/BF02671749
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DOI: https://doi.org/10.1007/BF02671749