On 2-Closedness of the Rational Numbers in Quasivarieties of Nilpotent Groups
The dominion of a subgroup H of a group G in a class M is the set of all elements a ∈ G that have equal images under every pair of homomorphisms from G to a group of M coinciding on H. A group H is said to be n-closed in M if for every group G = gr(H, a1,..., a n ) of M that contains H and is generated modulo H by some n elements, the dominion of H in G (in M) is equal to H. We prove that the additive group of the rational numbers is 2-closed in every quasivariety M of torsion-free nilpotent groups of class at most 3 whenever every 2-generated group of M is relatively free.
Keywordsquasivariety nilpotent group additive group of the rational numbers dominion 2-closed group
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