Regular Group

• A permutation group acting on is semiregular if no non-identity element of fixes a point of .

  • If is semiregular, then all all orbits have length equal to .

• A permutation group is regular if it is semiregular and transitive.

  • If is regular on , then .
  • The group acts on itself regularly.

• (Godsil e3.8) A transitive Abelian permutation group is regular.

  • Proof: Let be a transitive Abelian permutation group. We show that is semi-regular. Let be fixed by . By transitivity, there exists such that for any , . Since is Abelian

    Thus which means . Therefore, is semi-regular and in combination with transitivity, regular.

Links §

• Algebraic Graph Theory by Godsil and Royle

• Group Action

• Symmetric Group