## Invariant properties of large subgroups in Abelian $$p$$-groups.(English)Zbl 1196.20060

Summary: Suppose $$G$$ is an Abelian $$p$$-group with a large subgroup $$L$$. It is proved that $$G$$ is (1) $$p^{\omega+n}$$-projective, $$n\in\mathbb{N}\cup\{0\}$$; (2) $$p^{\omega+1}$$-injective; (3) projectively thick; (4) an $$\omega$$-elongation of a totally projective $$p$$-group (respectively of a summable $$p$$-group) by an $$p^{\omega+n}$$-projective group, $$n\in\mathbb{N}\cup\{0\}$$, and their modifications, precisely when so is $$L$$. These statements enlarge results due to K. M. Benabdallah, B. J. Eisenstadt, J. M. Irwin, E. W. Poluianov [Acta Math. Acad. Sci. Hung. 21, 421-435 (1970; Zbl 0215.39804)] and due to the author [Proc. Indian Acad. Sci., Math. Sci. 114, No. 3, 225-233 (2004; Zbl 1062.20059)]. – Some related concepts are established as well.

### MSC:

 20K10 Torsion groups, primary groups and generalized primary groups 20K27 Subgroups of abelian groups

### Keywords:

large subgroups; Abelian $$p$$-groups; projective groups

### Citations:

Zbl 0215.39804; Zbl 1062.20059