Cycle Graph (algebra)

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.