Cycle Graph (algebra)
In group theory, a sub-field of abstract algebra, a group cycle graph illustrates the various cycles of a group and is particularly useful in visualizing the structure of small finite groups. For groups with fewer than 16 elements, the cycle graph determines the group (up to isomorphism).
A cycle is the set of powers of a given group element a; where an, the n-th power of an element a, is defined as the product of a multiplied by itself n times. The element a is said to generate the cycle. In a finite group, some non-zero power of a must be the group identity, e; the lowest such power is the order of the cycle, the number of distinct elements in it. In a cycle graph, the cycle is represented as a series of polygons, with the vertices representing the group elements, and the connecting lines indicating that all elements in that polygon are members of the same cycle.
Read more about Cycle Graph (algebra): Cycles, Properties, Other Information Derivable From Cycle Graphs, Graph Characteristics of Particular Group Families
Famous quotes containing the words cycle and/or graph:
“The Buddha, the Godhead, resides quite as comfortably in the circuits of a digital computer or the gears of a cycle transmission as he does at the top of a mountain or in the petals of a flower.”
—Robert M. Pirsig (b. 1928)
“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)