Functions of the form or , where is a group, arise in lots of contexts.
One very natural way this can happen is to have a probability distribution on a group, . The probability density of group elements is a function .
Another way this can happen is if you have some function and has a natural action on ‘s domain – if you care about the values takes at a particular point , you are led to consider functions of the form . For a specific example, the intensity of a particular pixel, , in a square gray-scale image, , subject to flips and rotations, can be considered as a function .
Recall that we can visualize finitely generated groups by drawing Cayley Diagrams. (There’s a nice book, Visual Group Theory by Nathan Carter, that teaches a lot of basic group theory from the perspective of Cayley Diagrams.)
The natural way to visualize functions on groups is to picture them as taking values on the nodes of the Cayley Diagrams. One way to do this is by coloring the nodes. In the following visualization of a real-valued function on , dark colors represent a value being close to zero and light colors close to one.