Math Help - Cayley Diagram for $A_{4}$ with 12 elements - Carter Ex 4.6(c) page 54

1. Cayley Diagram for A4 with 12 elements - Carter Ex 4.6(c) page 54

I am a math novice and hobbyist reading Nathan Carter's book "Visual Group Theory"
I am currently trying to understand Cayley Diagrams, and in particular how to go formally from the Cayley Diagram to a multiplication table.

On page 54 of his book Nathan Carter gives the following diagram for $A_{4}$, the Alternating Group with 12 elements. I am taking the generator associated with the solid lines to be 'a', since there is a solid link from 'e' to 'a' , and similarly the dotted bidirectional lines to be the generator 'x' because of the link from 'e' to 'x'.

Some of the entries in the multiplication table are easy to read off the diagram - such as x*a = b because of the solid link from x to b.

However, how do you formally derive x*c fromthe diagram?

Bernhard

2. Originally Posted by Bernhard
I am a math novice and hobbyist reading Nathan Carter's book "Visual Group Theory"
I am currently trying to understand Cayley Diagrams, and in particular how to go formally from the Cayley Diagram to a multiplication table.

On page 54 of his book Nathan Carter gives the following diagram for $A_{4}$, the Alternating Group with 12 elements. I am taking the generator associated with the solid lines to be 'a', since there is a solid link from 'e' to 'a' , and similarly the dotted bidirectional lines to be the generator 'x' because of the link from 'e' to 'x'.

Some of the entries in the multiplication table are easy to read off the diagram - such as x*a = b because of the solid link from x to b.

However, how do you formally derive x*c fromthe diagram?

Bernhard
You would use the fact that c=ab (go from e to c via the shortest path). Then xc=xab=b^2.

However, cayley diagrams (also, perhaps more commonly, called cayley graphs) have a different niche from multiplication tables. In fact, multiplication tables are very rarely used, while cayley graphs are widely studied! Multiplication tables can only ever be used to look at finite groups, while cayley graphs are often used to study infinite groups (for example, look up `ends of a group').