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

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