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.


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