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Monothetic algebraic groups

Part of: Algebraic groups Linear algebraic groups and related topics Abelian groups

Published online by Cambridge University Press:  09 April 2009

Giovanni Falcone ,
Peter Plaumann  and
Karl Strambach
Giovanni Falcone
Affiliation:
Dipartimento di Metodi e Modelli MatematiciUniversità di Palermoviale delle Scienze1-90128 [email protected]
Peter Plaumann
Affiliation:
Mathematisches InstitutUniversität ErlangenBismarckstraße 1 ½D-91054 [email protected]@mi.uni-erlangen.de
Karl Strambach
Affiliation:
Mathematisches InstitutUniversität ErlangenBismarckstraße 1 ½D-91054 [email protected]@mi.uni-erlangen.de
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Abstract

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We call an algebraic group monothetic if it possesses a dense cyclic subgroup. For an arbitrary field k we describe the structure of all, not necessarily affine, monothetic k-groups G and determine in which cases G has a k-rational generator.

MSC classification

Secondary: 14L99: None of the above, but in this section 20G15: Linear algebraic groups over arbitrary fields 20K99: None of the above, but in this section
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 82 , Issue 3 , June 2007 , pp. 315 - 324
Copyright
Copyright © Australian Mathematical Society 2007

References

[1]Borel, A., Linear algebraic groups, Graduate Texts in Mathematics 126 (Springer-Verlag, New York, 1991).CrossRefGoogle Scholar
[2]Frey, G. and Jarden, M., ‘Approximation theory and the rank of abelian varieties over large algebraic fields’, Proc. Lond. Math. Soc. (3) 28 (1974), 112–128.CrossRefGoogle Scholar
[3]Fried, M. and Jarden, M., Field Arithmetic (Springer-Verlag, Berlin, 1986).CrossRefGoogle Scholar
[4]Hewitt, E. and Ross, K. A., Abstract harmonic analysis. Vol. I: Structure of topological groups. Integration theory. Group representations (Springer-Verlag, Berlin, 1963).Google Scholar
[5] J. Humphreys, E., Linear algebraic groups, Graduate Texts in Mathematics 21 (Springer-Verlag, New York, 1975, 1981).CrossRefGoogle Scholar
[6]Husemöller, D., Elliptic curves, Graduate Texts in Mathematics 111 (Springer-Verlag, New York, 1987).CrossRefGoogle Scholar
[7]Lang, S., ‘Number theory III: Diophantine geometry’, Encyclopaedia of Mathematical Sciences 60 (ed. Gamkrelize, R. V.) (Springer-Verlag, Berlin, 1991).CrossRefGoogle Scholar
[8]Mumford, D., Abelian varieties (Oxford University Press, Oxford, 1974).Google Scholar
[9]Plaumann, P., Strambach, K. and Zacher, G., ‘Der Verband der zusammenhängenden Untergruppen einer kommutativen algebraischen Gruppe’, Arch. Math. (Basel) 85 (2005), 37–48.CrossRefGoogle Scholar
[10]van der Put, M. and Singer, M. F., Galois theory of linear differential equations, Grundlehren der Mathematischen Wissenschaften 328 (Springer-Verlag, Berlin, 2003).CrossRefGoogle Scholar
[11]Rosenlicht, M., ‘Some basic theorems on algebraic groups’, Amer. J. Math. 78 (1956), 401–443.CrossRefGoogle Scholar
[12]Rosenlicht, M., ‘Extensions of vector groups by abelian vareties’ Amer. J. Math. 80 (1958), 685–714.CrossRefGoogle Scholar
[13]Rosenlicht, M., ‘The definition of field of definition’, Bol. Soc. Mat. Mexicana (2) 7 (1962), 39–46.Google Scholar
[14]van Dantzig, D., ‘Über topologisch homogene Kontinua’, Fund. Math. 15 (1930), 102–125.CrossRefGoogle Scholar
[15]Weil, A., Variétés abéliennes et courbes algébriques (Hermann & Cie, Paris, 1948).Google Scholar
[16]Weil, A., Foundations of algebraic geometry (American Mathematical Society, Providence, RI, 1962).Google Scholar