Abstract
Let G be a reduced abelian p-group. In a rare blemish in Kaplansky’s monograph Infinite abelian groups it is stated that the supremum of a finite number of U-sequences of G is taken pointwise. We provide an algorithm to show how the supremum of an arbitrary set of U-sequences should be calculated and use it to show that the lattice of U-sequences is distributive. This enables us to correct the proof of Kaplansky’s result that, when G is fully transitive, its lattice of fully invariant subgroups is distributive. We also prove, even when G is not fully transitive, that its lattice of large subgroups is distributive and we extend many of these results to non-reduced groups.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
B.A. Davey, H.A. Priestley, Introduction to Lattices and Order, 2nd edn. (Cambridge University Press, Cambridge, 2002)
L. Fuchs, Infinite Abelian Groups, vols. I & II (Academic, New York, 1973)
S.Y. Grinshpon, P.A. Krylov, Fully invariant subgroups, full transitivity and homomorphism groups of abelian groups. J. Math. Sci. 128, 2894–2997 (2005)
I. Kaplansky, Infinite Abelian Groups (The University of Michigan Press, Ann Arbor, 1954 & 1969)
R.S. Pierce, Homomorphisms of primary abelian groups, in Topics in Abelian Groups, ed. by J.M. Irwin, E.A.Walker (Scott, Foresman & Co, Chicago, 1963)
Acknowledgements
It is a pleasure to thank Brendan Goldsmith for his stimulus and encouragement. He kindly suggested the simple form of Algorithm A and the proof of Lemma 4.1. I am also grateful to the referee for providing a proof of Theorem 3.1 that is shorter than my original version and shows more clearly why the result is true.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this chapter
Cite this chapter
McLean, K.R. (2017). The Lattice of U-Sequences of an Abelian p-Group. In: Droste, M., Fuchs, L., Goldsmith, B., Strüngmann, L. (eds) Groups, Modules, and Model Theory - Surveys and Recent Developments . Springer, Cham. https://doi.org/10.1007/978-3-319-51718-6_24
Download citation
DOI: https://doi.org/10.1007/978-3-319-51718-6_24
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-51717-9
Online ISBN: 978-3-319-51718-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)