×

On variations of \(m,n\)-simply presented Abelian \(p\)-groups. (English) Zbl 1309.20046

This paper concerns generalizations of well known properties of infinite abelian \(p\)-groups, for example simply-presented or \(p^\lambda\)-projective. For a given property \(P\), a group \(G\) is called an \(n\)-\(P\)-group if \(G\) has a \(p^n\)-bounded subgroup \(H\) such that \(G/H\) is a \(P\)-group; and \(G\) is an \(m,n\)-\(P\)-group if there exists an \(m\)-\(P\)-group \(H\) containing a \(p^n\)-bounded subgroup \(K\) such that \(G=H/K\). The paper contains a host of variations of these definitions and considers the relations among these properties as \(m,n\) and \(P\) vary. The author considers also the inheritance of the properties under the operations \(G\to p^\lambda G\) and \(G\to G/p^\lambda G\) for various ordinals \(\lambda\). He exhaustively settles these problems or poses them as Conjectures or Questions.

MSC:

20K10 Torsion groups, primary groups and generalized primary groups
20K40 Homological and categorical methods for abelian groups
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Danchev P. A note on m, n-balanced projective abelian p-groups. J Algebra Number Theory Acad, 2013, 3: 203-207
[2] Danchev P. On strongly and separably ω1-pω+n-projective abelian p-groups. Hacettepe J Math Stat, 2014, 43: 51-64 · Zbl 1312.20048
[3] Danchev, P., On m-ω1-pω+n-projective abelian p-groups (2015) · Zbl 1339.20046
[4] Danchev P, Keef P. An application of set theory to ω + n-totally pω+n-projective primary abelian groups. Mediterr J Math, 2011, 8: 525-542 · Zbl 1247.20063 · doi:10.1007/s00009-010-0088-2
[5] Fuchs L. Infinite Abelian Groups. New York-London: Academic Press, 1970, 1973 · Zbl 0209.05503
[6] Griffith P. Infinite Abelian Group Theory. Chicago-London: The University of Chicago Press, 1970 · Zbl 0204.35001
[7] Keef P. On ω1-pω+n-projective primary abelian groups. J Algebra Number Theory Acad, 2010, 1: 41-75 · Zbl 1229.20055
[8] Keef P, Danchev P. On n-simply presented primary abelian groups. Houston J Math, 2012, 38: 1027-1050 · Zbl 1271.20066
[9] Keef P, Danchev P. On properties of n-totally projective abelian p-groups. Ukrain Math J, 2012, 64: 766-771 · Zbl 1259.20059
[10] Keef P, Danchev P. On m, n-balanced projective and m, n-totally pojective primary abelian groups. J Korean Math Soc, 2013, 50: 307-330 · Zbl 1273.20053 · doi:10.4134/JKMS.2013.50.2.307
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.