Dihedral Group - Small Dihedral Groups

Small Dihedral Groups

For n = 1 we have Dih1. This notation is rarely used except in the framework of the series, because it is equal to Z2. For n = 2 we have Dih2, the Klein four-group. Both are exceptional within the series:

  • They are abelian; for all other values of n the group Dihn is not abelian.
  • They are not subgroups of the symmetric group Sn, corresponding to the fact that 2n > n ! for these n.

The cycle graphs of dihedral groups consist of an n-element cycle and n 2-element cycles. The dark vertex in the cycle graphs below of various dihedral groups stand for the identity element, and the other vertices are the other elements of the group. A cycle consists of successive powers of either of the elements connected to the identity element.

Dih1 Dih2 Dih3 Dih4 Dih5 Dih6 Dih7

Read more about this topic:  Dihedral Group

Famous quotes containing the words small and/or groups:

    The great disadvantage, and advantage, of the small urban bourgeois is his limited outlook. He sees the world as a middle- class world, and everything outside these limits is either laughable or slightly wicked.
    George Orwell (1903–1950)

    In properly organized groups no faith is required; what is required is simply a little trust and even that only for a little while, for the sooner a man begins to verify all he hears the better it is for him.
    George Gurdjieff (c. 1877–1949)