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)
“The conflict between the need to belong to a group and the need to be seen as unique and individual is the dominant struggle of adolescence.”
—Jeanne Elium (20th century)
“Being dismantled before our eyes are not just individual programs that politicians cite as too expensive but the whole idea that society has a stake in the well-being of children down the block and the security of families on the other side of town. Whether or not kids eat well, are nurtured and have a roof over their heads is not just a consequence of how their parents behave. It is also a responsibility of society—but now apparently a diminishing one.”
—Richard B. Stolley (20th century)