Advertisement

Mediterranean Journal of Mathematics

, Volume 8, Issue 3, pp 307–313 | Cite as

Periodic Endomorphisms of Polycyclic Groups

  • Gerard Endimioni
Article

Abstract

Let G be a polycyclic group. As a consequence of known results, any periodic group of automorphisms of G is finite and there is an upper bound (depending only on G) for its order. On the other hand, a periodic semigroup of endomorphisms of G need not be finite but we prove that it is locally finite. Also we show that the order of periodic endomorphisms of G is bounded.

Mathematics Subject Classification (2010)

20E15 20M20 

Keywords

Polycyclic group periodic endomorphism periodic semigroup 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    A. H. Clifford and G. B. Preston, The algebraic theory of semigroups, Math. Survey, no. 7, Amer. Math. Soc., Providence, R. I., 1961.Google Scholar
  2. 2.
    C. W. Curtis and I. Reiner, Representation theory of finite groups and associative algebras, Pure and Appl. Math., vol. 11, Interscience, New York, 1962.Google Scholar
  3. 3.
    I. Kaplansky Fields and ring, 2nd ed. The University of Chicago Press, Chicago and London (1972)Google Scholar
  4. 4.
    Kuzmanovich J., Pavlichenkov A.: Finite groups of matrices whose entries are integers. Amer. Math. Monthly 109, 173–186 (2002)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    McNaughton R., Zalcstein Y.: The Burnside problem for semigroups. J. Algebra 34, 292–299 (1975)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Merzljakov Ju. I.: Integral representation of the holomorphs of polycyclic groups. Algebra i Logika 9, 539–558 (1970) (Russian). English transl.: Algebra and Logic 9 (1970), 326-337MathSciNetGoogle Scholar

Copyright information

© Springer Basel AG 2010

Authors and Affiliations

  1. 1.C.M.I.–Université de ProvenceMarseille Cedex 13France

Personalised recommendations