On Abelian Groups Having All Proper Fully Invariant Subgroups Isomorphic

Abstract

We introduce two classes of abelian groups which have either only trivial fully invariant subgroups or all their nontrivial (respectively nonzero) fully invariant subgroups are isomorphic, called IFI-groups and strongly IFI-groups, such that every strongly IFI-group is an IFI-group, respectively. Moreover, these classes coincide when the groups are torsion-free, but are different when the groups are torsion as well as, surprisingly, mixed groups cannot be IFI-groups. We also study their important properties as our results somewhat contrast with those from [13] and [14].

Key Words:

• Abelian groups
• Characteristic subgroups
• Fully invariant subgroups
• Projection invariant subgroups

2010 Mathematics Subject Classification:

• Primary: 20K10, 20K12

ACKNOWLEDGMENT

The authors would like to thank the referee for the thoughtful comments on the article and to thank the editor, Professor A. Olshanskii, for his efforts and patience in processing this work.

Notes

Communicated by A. Olshanskii.