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Semigroup Forum

, Volume 87, Issue 2, pp 467–488 | Cite as

On commutation semigroups of dihedral groups

  • Darien DeWolf
  • Charles Edmunds
  • Christopher Levy
RESEARCH ARTICLE
  • 158 Downloads

Abstract

For any group G with gG, the right and left commutation semigroups associated with g are the mappings ρ(g) and λ(g) from G to G defined as (x)ρ(g)=[x,g] and (x)λ(g)=[g,x]. The set M(G) of all mappings from G to G forms a semigroup under composition of mappings. The right and left commutation semigroups of G, denoted P(G) and Λ(G), are the subsemigroups of M(G) generated by {ρ(g):gG} and {λ(g):gG}, respectively. In this paper, we develop explicit formulas for the orders of P(G) and Λ(G) when G=D m , the dihedral group of order 2m. We apply these formulas to address the problems of determining when |P(G)|=|Λ(G)| and P(G)≅Λ(G) and to derive proofs of several previous results of James Countryman (Ph.D. Thesis, University of Notre Dame, 1970) on commutation semigroups of pq groups.

Keywords

Commutation semigroup Dihedral groups 

References

  1. 1.
    Countryman, J.J.: On the commutation semigroups of pq groups. Ph.D. Thesis, University of Notre Dame (1970), p. 58 Google Scholar
  2. 2.
    DeWolf, D.: Commutation semigroups of dihedral groups of order 2n where n is even. Honours Thesis, Mount Saint Vincent University (2012), p. 20 Google Scholar
  3. 3.
    Gupta, N.D.: On commutation semigroups of a group. J. Aust. Math. Soc. 6, 36–45 (1966) CrossRefzbMATHGoogle Scholar
  4. 4.
    Gupta, N.D.: Commutation near-rings of a group. J. Aust. Math. Soc. 7, 135–140 (1967) CrossRefzbMATHGoogle Scholar
  5. 5.
    Levy, C.D.: Investigation of commutation semigroups of dihedral groups. Honours Thesis, Mount Saint Vincent University (2009), p. 26 Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Darien DeWolf
    • 1
  • Charles Edmunds
    • 2
  • Christopher Levy
    • 1
  1. 1.Mathematics DepartmentDalhousie UniversityHalifaxCanada
  2. 2.Mathematics DepartmentMount Saint Vincent UniversityHalifaxCanada

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