Dihedral Group - Automorphism Group

Automorphism Group

The automorphism group of Dihn is isomorphic to the affine group Aff(Z/nZ) and has order where is Euler's totient function, the number of k in coprime to n.

It can be understood in terms of the generators of a reflection and an elementary rotation (rotation by, for k coprime to n); which automorphisms are inner and outer depends on the parity of n.

  • For n odd, the dihedral group is centerless, so any element defines a non-trivial inner automorphism; for n even, the rotation by 180° (reflection through the origin) is the non-trivial element of the center.
  • Thus for n odd, the inner automorphism group has order 2n, and for n even the inner automorphism group has order n.
  • For n odd, all reflections are conjugate; for n even, they fall into two classes (those through two vertices and those through two faces), related by an outer automorphism, which can be represented by rotation by (half the minimal rotation).
  • The rotations are a normal subgroup; conjugation by a reflection changes the sign (direction) of the rotation, but otherwise leaves them unchanged. Thus automorphisms that multiply angles by k (coprime to n) are outer unless

Read more about this topic:  Dihedral Group

Famous quotes containing the word group:

    “around our group I could hear the wilderness listen.”
    —William Stafford (1914–1941)