Torsion Groups with the Norm of pd-Subgroup of Finite Index

T.D. Lukashova; M.G. Drushlyak; A.V. Pidopryhora

doi:10.15421/242410

Torsion Groups with the Norm of pd-Subgroup of Finite Index

T.D. Lukashova (Sumy State Pedagogical University named after A.S. Makarenko)
M.G. Drushlyak (Sumy State Pedagogical University named after A.S. Makarenko), https://orcid.org/0000-0002-9648-2248
A.V. Pidopryhora (Sumy State Pedagogical University named after A.S. Makarenko)

Abstract

The authors study the relations between the properties of torsion groups and their norms of $$$pd$$$-subgroups. The norm $$$N_G^{pdI}$$$ of $$$pd$$$-subgroups of a group $$$G$$$ is the intersection of the normalizers of all its $$$pd$$$-subgroups or a group itself, if the set of such subgroups is empty in a group. The structure of the norm of $$$pd$$$-subgroups in torsion groups is described and the conditions of Dedekindness of this norm is proved (Dedekind group is a group in which all subgroups are normal). It is proved that a torsion group is a finite extension of its norm of $$$pd$$$-subgroups if and only if it is a finite extension of its center. By this fact and the structure of the norm of $$$pd$$$-subgroups, we get that any torsion group that is a finite extension of this norm is locally finite.

Keywords

torsion group; normalizer; norm of a group; norm of $$$pd$$$-subgroups; non-Dedekindness; subgroup index in a group; center of a group

MSC 2020

20E07

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