Abstract
The dominion of a subgroup H in a group G (in the class of metabelian groups) is the set of all elements a ∈ G whose images are equal for all pairs of homomorphisms from G into every metabelian group that coincide on H. The dominion is a closure operator on the lattice of subgroups of G. We study the closed subgroups with respect to the dominion. It is proved that if G is a metabelian group, H is a locally cyclic group, the commutant G′ of G is the direct product of its subgroups of the form H f (f ∈ G), and G′ = H G × K for a suitable subgroup K; then the dominion of H in G coincides with H.
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Original Russian Text Copyright © 2014 Budkin A.I.
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Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 55, No. 6, pp. 1240–1249, November–December, 2014.
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Budkin, A.I. On the closedness of a locally cyclic subgroup in a metabelian group. Sib Math J 55, 1009–1016 (2014). https://doi.org/10.1134/S0037446614060044
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DOI: https://doi.org/10.1134/S0037446614060044