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.
|
Read more about this topic: Dihedral Group
Famous quotes containing the words small and/or groups:
“Enthusiasm. A distemper of youth, curable by small doses of repentance in connection with outward applications of experience.”
—Ambrose Bierce (1842–1914)
“Some of the greatest and most lasting effects of genuine oratory have gone forth from secluded lecture desks into the hearts of quiet groups of students.”
—Woodrow Wilson (1856–1924)