Group Explorer 3.0 Help: Cayley Diagram Overview

Introduction to Cayley diagrams

The Cayley diagram is the clearest visualizer for depicting groups as sets with structure. They also clearly show the group as acting on itself. If you have not yet encountered the multiplication table visualizer, start with these help pages:

• The definition of a Cayley diagram
• The documentation on the Cayley diagram interface in Group Explorer

CITE(VGT-2.4 MM-1.2 DE-3)

What Cayley diagrams do well

The Cayley diagram is the only visualizer in Group Explorer with the following two useful properties.

• Each element in the group is represented by a unique item in the visualizer.
• Full information about the group structure is immediately visible in the diagram.

Cycle graphs also satisfy the first of these requirements but not the second. Multiplication tables satisfy the second but not the first.

Consider the Cayley diagram of $S_3$ shown below. There is a node in the diagram for each element of the group and all information for determining the structure of the group can be immediately seen (or sometimes inferred) from the arrows connecting the nodes.

The fact that a red arrow goes from $f$ to $fr$ indicates that the red arrows signify right-multiplication by the generator $r$. So one can see how two generators (the red arrows meaning $r$ and the blue-green arcs meaning $f$) combine to create all the structure of the group. Following a red arrow and then a blue-green one will lead you to a different location than following a blue-green arrow and then a red one; in this way one can see that $S_3$ is not abelian.

Some groups in Group Explorer come with a human-designed Cayley diagram that has a particular beauty or symmetry which best shows off the symmetry inherent in the group. Such diagrams are included if the standard algorithms Group Explorer uses for auto-generating Cayley diagrams do not yield aesthetically good results for that group. Here are two particularly pleasing Cayley diagrams.

A Cayley diagram for the group $S_4$

Node color is highlighted to indicate membership in cosets of the subgroup $\langle(1~2~3)\rangle$.

A Cayley diagram for the group $A_5$

Node color is highlighted to indicate membership in cosets of the subgroup $\langle(0~1~2~3~4)\rangle$.

There are some illustrations can be made from just about any kind of visualization, including Cayley diagrams. For instance, although this lattice of subgroups and this short exact sequence of groups use multiplication tables, they could just as easily have used Cayley diagrams instead. In fact, you can ask Group Explorer to automatically generate such visualizations for you; see the help on sheets.