On a permutablity problem for groups

  • Bijan Taeri
Article

Abstract

Letm, n be positive integers. We denote byR(m, n) (respectivelyP(m, n)) the class of all groupsG such that, for everyn subsetsX1, X2, . . .,Xn of sizem ofG there exits a non-identity permutation σ such that\(X_1 X_2 ...X_n \cap X_{\sigma (1)} X_{\sigma (2)} ...X_{\sigma (n)} \ne \not 0\) (respectively X1X2 . . .Xn = Xσ(1)X{σ(2)} . . . X{gs(n)}). Let G be a non-abelian group. In this paper we prove that
  1. (i)

    G ∈ P(2,3) if and only ifG isomorphic to S3, whereSn is the symmetric group onn letters.

     
  2. (ii)

    G ∈ R(2, 2) if and only if¦G¦ ≤ 8.

     
  3. (iii)

    IfG is finite, thenG ∈ R(3, 2) if and only if¦G¦ ≤ 14 orG is isomorphic to one of the following: SmallGroup(16,i), i ∈ {3, 4, 6, 11, 12, 13}, SmallGroup(32,49), SmallGroup(32, 50), where SmallGroup(m, n) is the nth group of orderm in the GAP [13] library.

     

AMS Mathematics Subject Classification

20F99 20D15 

Key words and phrases

Permutable group finite groups GAP 

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Copyright information

© Korean Society for Computational & Applied Mathematics and Korean SIGCAM 2006

Authors and Affiliations

  • Bijan Taeri
    • 1
  1. 1.Department of Mathematical SciencesIsfahan University of TechnologyIsfahanIran

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