There's a math problem in Pinter's Abstract Algebra book on pp. 52(exercise G):
Every finite group may be represented by a diagram known as a Cayley diagram. A Cayley diagram consists of points
joined by arrows.
There is one point for every element of the group.
The arrows represent the result of multiplying by a generator.
For example, if has only one generator (that is, is the cyclic group ), then the arrow
represents the operation "multiply by ":
If the group has two generators, say and , we need two kinds of arrows, say dotted arrow and
line arrow with no dots where dotted arrow means "multiply by " and lined arrow means "multiply by ".
For example, the group where and has the following
Cayley diagram(Figure 1):
Moving in the forward direction of the lined arrow means multiplying by ,
whereas moving in the backward direction of the lined arrow means multiplying by :
(Note that "multiplying by " understood to mean multiplying on the right by :
it means , not ) It is also a convention that if (hence ), then no
arrowhead is used:
for if , then multiplying by is the same as multiplying by
The Cayley diagram of a group contains the same information as the group's table. For instance, to find the product
in the previous figure 1, we start at and follow the path corresponding to (multiplying by , then
by , then again by ), which is (Figure 2)
This path leads to , hence
As another example, the inverse of is the path which leads from back to .
We note instantly that this is .
A point-and-arrow diagram is the Cayley diagram of a group iff it has the following two properties:
1) For each point and generator , there is exactly one -arrow ending at ;
furthermore, at most one arrow goes from to another point .
2) If two different paths starting at lead to the same destination, then those two paths, starting at any point , lead
to the same destination.
Cayley diagrams are a useful way of finding new groups.
Write the table of the groups having the following Cayley diagram (Remark: You may take any point to represent (neutral element), because
there is perfect symmetry in a Cayley diagram. Choose , then label the diagram and proceed.)
My problem is: Is it possible to kindly help me find this cyclic group's operation table? How do I choose the elements from this diagram?