In group theory a group cycle graph illustrates the various cycles of a group and is particularly useful in visualizing the structure of small finite groups. A cycle is a sequence of increasing powers of a particular group element where the n-th power of an element a is defined as the product of a multiplied by itself n times. Ultimately, a cycle always closes upon itself because there is always a value of n>0 for which an yields the identity element. 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 memebers of the same cycle.
Every cycle contains the identity element. Every finite group consists of a set of cycles, some of which may be separate, and some of which may be connected by a common element other than the identity element. To see that this is true, consider a partial sequence e, a, a2 ... an which is unclosed. The next element in the sequence an+1 must either be the identity element which closes the sequence to form a cycle, another member of the sequence which is not the identity element, or an element not yet in the sequence. The next element cannot be a non-identity member of the sequence because then an+1=am where m<n+1 which means that an+1-m=e which means the sequence has already been closed, which contradicts the original assumption that it is not. If the next element is the identity element, then the sequence closes to form a cycle, and if it is not then we add the new element to the sequence and continue the process of building the cycle.
Properties
As an example of a group cycle graph, consider the D8 group. The multiplication table for this group is shown on the left, and the cycle graph is shown on the right with e specifying the identity element.
| o | e | b | a | a2 | a3 | ab | a2b | a3b |
|---|
| e | e | b | a | a2 | a3 | ab | a2b | a3b |
|---|
| b | b | e | a3b | a2b | ab | a3 | a2 | a |
|---|
| a | a | ab | a2 | a3 | e | a2b | a3b | b |
|---|
| a2 | a2 | a2b | a3 | e | a | a3b | b | ab |
|---|
| a3 | a3 | a3b | e | a | a2 | b | ab | a2b |
|---|
| ab | ab | a | b | a3b | a2b | e | a3 | a2 |
|---|
| a2b | a2b | a2 | ab | b | a3b | a | e | a3 |
|---|
| a3b | a3b | a3 | a2b | ab | b | a2 | a | e |
|---|
Notice the cycle e, a, a2, a3 . It can be seen from the multiplication table that successive powers of a  in fact behave this way. The reverse case is also true. In other words: (a3)2=a2 , (a3)3=a and (a3)4=e . This behavior is true for any cycle in any group - a cycle may be traversed in either direction.
Cycles which contain a non-prime number of elements will implicitly have cycles which are not connected in the graph. For the D8 group above, we might want to draw a line between a2 and e since (a2)2=e but since a2 is part of a larger cycle, this is not done.
There can be ambiguity when two cycles share an element which is not the identity element. Consider for example, the simple quaternion group, whose cycle graph is shown on the right. Each of the elements in the middle row when multiplied by itself gives -1 (where 1 is the identity element). In this case we may use different colors to keep track of the cycles, although symmetry considerations will work as well.
Strictly speaking, the 2-element cycles should be connected by two lines, but this is usually abbreviated by a single line.
Two distinct groups may have cycle graphs that have the same structure, and can only be distinguished by the product table, or by labeling the elements in the graph in terms of
the groups basic elements. The lowest order for which this problem can occur is order 16 in the case of C2 x C8 and the modular group, as shown below. (Note - the cycles with common elements are distinguished by symmetry in these graphs.)
Other information derivable from cycle graphs
- The inverse of any element can be located on the cycle graph. The inverse of an element is simply its opposite element in the same cycle. If a cycle contains an odd number of elements then there will be one element which is its own opposite, and therefore is its own inverse. For example, in the D8 graph above, the inverse of a is its opposite, a3, while a2 is its own inverse.
Graph characteristics of particular group families
Certain group types give typical graphs:
- Cyclic groups Cn is a single cycle graphed simply as an n-sided polygon with the elements at the vertices.
- Groups of the form (Cn)m will have (nm-1)/(n-1) n-element cycles sharing the common identity element.
- Symmetry groups - Every symmetry group Sn contains as a subgroup all groups of order n. Thus the graph of every group of order n will be found in a graph of a symmetry group.
See also
List of small groups
References
- Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, 1993.