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How can I retrieve operation table of group elements from this Cayley diagram?
There's a math problem in Pinter's Abstract Algebra book on pp. 52(exercise G):
Exercise:
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):
Attachment 25767
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 (ab^2))
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)
Attachment 25766
This path leads to , hence (ab^2) = b)
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.
Problem:
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.)
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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?
Re: How to easily retrieve operation table of group elements from this Cayley diagram
Hey x3bnm.
Consider that the Cayley graph tells how you multiply one element by another returns a mapped result indicated by the node, the edge, and the arrow associated with those.
Just take that information and put it into a table and that is your group operation table noting the order of the multiplication.
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Re: How to easily retrieve operation table of group elements from this Cayley diagram
