Danchev, Peter V. Characteristic properties of large subgroups in primary Abelian groups. (English) Zbl 1062.20059 Proc. Indian Acad. Sci., Math. Sci. 114, No. 3, 225-233 (2004). The purpose of this paper is the study of relations between the structures of primary Abelian groups and their large subgroups. The results have the form “\(G\) belongs to the class of Abelian \(p\)-groups \(\mathcal K\) if and only if a fixed large subgroup \(L\) belongs to \(\mathcal K\)”. This is proved to hold for several classes of primary Abelian groups: summable, \(\sigma\)-summable, \(\Sigma\)-groups, \(p^{\omega+1}\)-projective. Reviewer’s remark: Some similar results were already proved in the 70’s for the standard example of large subgroups \(L=p^nG\). Reviewer: Grigore Călugăreanu (Safat) Cited in 1 ReviewCited in 5 Documents MSC: 20K10 Torsion groups, primary groups and generalized primary groups 20K27 Subgroups of abelian groups Keywords:large subgroups; Abelian \(p\)-groups; summable \(p\)-groups; \(\sigma\)-summable groups; projective groups; pure-complete \(p\)-groups PDF BibTeX XML Cite \textit{P. V. Danchev}, Proc. Indian Acad. Sci., Math. Sci. 114, No. 3, 225--233 (2004; Zbl 1062.20059) Full Text: DOI arXiv OpenURL References: [1] Benabdallah K M, Eisenstadt B J, Irwin J M and Poluianov E W, The structure of large subgroups of primary abelian groups,Acta Math. Acad, Sci. Hungaricae 21(3-4) (1970) 421–435 · Zbl 0215.39804 [2] Cutler D O, Quasi-isomorphism for infinite abelianp-groups,Pacific J. Math. 16(1) (1966) 25–45 · Zbl 0136.28904 [3] Danchev P V, Commutative group algebras of abelian {\(\Sigma\)}-groups,Math. J. Okayama Univ. 40(2) (1998) 77–90 · Zbl 0953.16024 [4] Danchev P V, Theorems of the type of Nunke for abelian groups,Compt. rend. Acad. bulg. Sci. 53(10) (2000) 5–8 · Zbl 0964.20030 [5] Danchev P V, Generalized Dieudonné criterion (to appear) [6] Fuchs L, Infinite Abelian Groups, I and II (Moscow, Mir) (1974 and 1977) (in Russian) · Zbl 0274.20067 [7] Hill P D, On the structure of abelianp-groups,Trans. Am. Math. Soc. 288 (1985) 505–525 · Zbl 0573.20053 [8] Khan M Z, Large and high subgroups,Proc. Indian Acad, Sci. Sect. A 87(9) (1978) 177–179 · Zbl 0412.20048 [9] Nunke R J, Purity and subfunctors of identity, Topics in abelian groups,Proc. Symposium, New Mexico State University (1962) pp. 121–171; (Chicago, Illinois: Scott Foresman and Co.) (1963) [10] Richman F, Thin abelian p-groups,Pacific J. Math. 27(3) (1968) 599–606 · Zbl 0181.03702 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.