×

Generalizations of fully transitive and valuated abelian \(p\)-groups. (English) Zbl 1473.20056

All groups are additively written abelian \(p\)-groups. The authors generalize to the class of valuation groups the notion of fully transitive group introduced by Kaplansky. Let \(H\) be a subgroup of an abelian \(p\)-group \(G\), then \(G\) is called \(H\)-fully transitive if using the height valuation from \(G\), for every \(x \in H\), every valuated (i.e., non-height decreasing) homomorphism \(\langle x\rangle\rightarrow G\) extends to a valuated homomorphism \(H \rightarrow G\). The authors prove a number of statements in which \(H\)-fully transitivity implies fully transitivity. They construct the example group \(G\) such that \(G\) is \(H\)-fully transitive but \(G\) is itself not fully transitive. The property \(H\)-fully transitivity is researched in the context of direct sums and direct products. The authors characterize those valuated groups \(H\) that are universally fully transitive in the sense that every group \(G\) that contains \(H\) as such an embedded subgroup is necessarily \(H\)-fully transitive. Some properties of the class of universally fully transitive groups are discussed. The authors show that the class of universally fully transitive groups form the smallest class of groups containing the totally projective groups that are closed under \(t\)-products. It is proved that if \(G\) and \(H\) are universally fully transitive, then so is \(\mathrm{Tor}(G, H)\). Some open problems are formulated.

MSC:

20K10 Torsion groups, primary groups and generalized primary groups
20K30 Automorphisms, homomorphisms, endomorphisms, etc. for abelian groups
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Corner, A. L.S., On endomorphism rings of primary Abelian groups II, Q. J. Math. Oxf., 27, 5-13 (1976) · Zbl 0326.20047
[2] Danchev, P. V.; Goldsmith, B., On socle regularity and some notions of transitivity for abelian p-groups, J. Commut. Algebra, 3, 3, 301-319 (2011) · Zbl 1247.20062
[3] Fuchs, L., Abelian Groups (1958), Publ. House of the Hungar. Acad. Sci: Publ. House of the Hungar. Acad. Sci Budapest · Zbl 0090.02003
[4] Fuchs, L., Infinite Abelian Groups, vol. I (1970), Acad. Press: Acad. Press New York and London; Fuchs, L., Infinite Abelian Groups, vol. II (1973), Acad. Press: Acad. Press New York and London · Zbl 0209.05503
[5] Fuchs, L., Vector spaces with valuations, J. Algebra, 35, 23-38 (1975) · Zbl 0318.15002
[6] Fuchs, L., Abelian Groups (2015), Springer: Springer Switzerland · Zbl 1416.20001
[7] Fuchs, L.; Irwin, J., On \(p^{\omega + 1}\)-projective p-groups, Proc. Lond. Math. Soc., 30, 459-470 (1975) · Zbl 0324.20059
[8] Goldsmith, B.; Strüngmann, L., Some transitivity results for torsion abelian groups, Houst. J. Math., 23, 941-957 (2007) · Zbl 1139.20052
[9] Griffith, P., Transitive and fully transitive primary abelian groups, Pac. J. Math., 25, 249-254 (1968) · Zbl 0157.05901
[10] Griffith, P., Infinite Abelian Group Theory (1970), The University of Chicago Press: The University of Chicago Press Chicago and London · Zbl 0204.35001
[11] Hennecke, G., Transitivity and full transitivity over subgroups of abelian p-groups, (Abelian Groups and Modules. Abelian Groups and Modules, Trends in Math. (1999), Birkhäuser Verlag: Birkhäuser Verlag Basel), 43-53 · Zbl 0940.20053
[12] Hill, P., On transitive and fully transitive primary groups, Proc. Am. Math. Soc., 22, 414-417 (1969) · Zbl 0192.35601
[13] Hill, P., On the structure of abelian p-groups, Trans. Am. Math. Soc., 288, 505-525 (1985) · Zbl 0573.20053
[14] Hill, P.; Megibben, C., On direct sums of countable groups and generalizations, (Studies on Abelian Groups (1968), Springer: Springer Berlin), 183-206 · Zbl 0203.32705
[15] Kaplansky, I., Infinite Abelian Groups (1954), University of Michigan Press: University of Michigan Press Ann Arbor, and 1969 · Zbl 0057.01901
[16] Keef, P., On the Tor functor and some classes of abelian groups, Pac. J. Math., 132, 63-84 (1988) · Zbl 0617.20033
[17] Richman, F.; Walker, E. A., Valuated groups, J. Algebra, 56, 1, 145-167 (1979) · Zbl 0401.20049
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.