Archiv der Mathematik

, 97:209 | Cite as

Finite groups are big as semigroups

  • I. Dolinka
  • N. Ruškuc


We prove that a finite group G occurs as a maximal proper subsemigroup of an infinite semigroup (in the terminology of Freese, Ježek, and Nation, G is a big semigroup) if and only if |G| ≥ 3. In fact, any finite semigroup whose minimal ideal contains a subgroup with at least three elements is big.

Mathematics Subject Classification (2010)

Primary 20M10 Secondary 20F05 20F50 


Finite maximal subsemigroup Rees matrix semigroup 


  1. 1.
    S. I. Adyan and I. G. Lysionok, On groups all of whose proper subgroups are finite cyclic, Izv. Akad. Nauk SSSR, Ser. Mat. 55 (1991), 933–990 (in Russian; English transl. Math. USSR-Izv. 39 (1992), 905–957).Google Scholar
  2. 2.
    Clifford A.H., Preston G.B.: Algebraic Theory of Semigroups, Vol I. American Mathematical Society, Providence (1961)zbMATHGoogle Scholar
  3. 3.
    Freese R., Ježek J., Nation J.B.: Lattices with large minimal extensions. Algebra Universalis 45, 221–309 (2001)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Graham N., Graham R., Rhodes J.: Maximal subsemigroups of finite semigroups. J. Comb. Theory 4, 203–209 (1968)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Howie J.M.: Fundamentals of Semigroup Theory. Oxford University Press, New York (1995)zbMATHGoogle Scholar
  6. 6.
    A. Yu. Ol’shanskiĭ, Groups of bonded period with subgroups of prime order, Algebra i Logika 21 (1982), 553–618 (in Russian; English transl. Algebra and Logic 21 (1982), 369–418).Google Scholar
  7. 7.
    Petrich M.: The translational hull of a completely 0-simple semigroup. Glasgow Math. J. 9, 1–11 (1968)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Department of Mathematics and InformaticsUniversity of Novi SadNovi SadSerbia
  2. 2.School of Mathematics and StatisticsUniversity of St AndrewsSt AndrewsScotland, UK

Personalised recommendations