Graph Characteristics of Particular Group Families
Certain group types give typical graphs:
- Cyclic groups Zn is a single cycle graphed simply as an n-sided polygon with the elements at the vertices.
| Z1 | Z2 | Z3 | Z4 | Z5 | Z6 | Z7 | Z8 |
|---|
- When n is a prime number, groups of the form (Zn)m will have (nm-1)/(n-1) n-element cycles sharing the common identity element.
| Z2² | Z2³ | Z24 | Z3² |
|---|
- Dihedral groups Dihn consists of an n-element cycle and n 2-element cycles.
| Dih1 | Dih2 | Dih3 | Dih4 | Dih5 | Dih6 | Dih7 |
|---|
- Symmetric groups - The symmetric group Sn contains, for any group of order n, a subgroup isomorphic to that group. Thus the cycle graph of every group of order n will be found in the cycle graph of Sn. See example: Subgroups of S4
Read more about this topic: Cycle Graph (algebra)
Famous quotes containing the words graph, group and/or families:
“In this Journal, my pen is a delicate needle point, tracing out a graph of temperament so as to show its daily fluctuations: grave and gay, up and down, lamentation and revelry, self-love and self-disgust. You get here all my thoughts and opinions, always irresponsible and often contradictory or mutually exclusive, all my moods and vapours, all the varying reactions to environment of this jelly which is I.”
—W.N.P. Barbellion (1889–1919)
“Stripped of ethical rationalizations and philosophical pretensions, a crime is anything that a group in power chooses to prohibit.”
—Freda Adler (b. 1934)
“We as a nation need to be reeducated about the necessary and sufficient conditions for making human beings human. We need to be reeducated not as parents—but as workers, neighbors, and friends; and as members of the organizations, committees, boards—and, especially, the informal networks that control our social institutions and thereby determine the conditions of life for our families and their children.”
—Urie Bronfenbrenner (b. 1917)