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Dominions in Solvable Groups

Research paper by A. I. Budkin

Indexed on: 10 Mar '16Published on: 05 Dec '15Published in: Algebra and Logic



Abstract

The dominion of a subgroup H of a group G in a class M is the set of all elements a ∈ G whose images are equal for all pairs of homomorphisms from G to each group in M that coincide on H. A group H is absolutely closed in a class M if, for any group G in M and any inclusion H ≤ G, the dominion of H in G (with respect to M) coincides with H (i.e., H is closed in G). We prove that every torsion-free nontrivial Abelian group is not absolutely closed in ANc. It is shown that if a subgroup H of G in NcA has trivial intersection with the commutator subgroup G′, then the dominion of H in G (with respect to NcA) coincides with H. It is stated that the study of closed subgroups reduces to treating dominions of finitely generated subgroups of finitely generated groups.