# How to easily retrieve operation table of group elements from this Cayley diagram?

• Nov 17th 2012, 02:30 PM
x3bnm
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 $G$ has only one generator $a$ (that is, $G$ is the cyclic group $\langle a \rangle$), then the arrow $\rightarrow$
represents the operation "multiply by $a$":

$e\rightarrow a \rightarrow a^2 \rightarrow a^3 \rightarrow \cdots$

If the group has two generators, say $a$ and $b$, we need two kinds of arrows, say dotted arrow and
line arrow with no dots where dotted arrow means "multiply by $a$" and lined arrow means "multiply by $b$".

For example, the group $G = \{ e, a, b, b^2, ab, ab^2 \}$ where $a^2 = e, b^3 = e,$ and $ba = ab^2$ has the following
Cayley diagram(Figure 1):

Attachment 25767

Moving in the forward direction of the lined arrow means multiplying by $b$,

$x \rightarrow xb$

whereas moving in the backward direction of the lined arrow means multiplying by $b^{-1}$:

$x \leftarrow xb^{-1}$

(Note that "multiplying $x$ by $b$" understood to mean multiplying on the right by $b$:
it means $xb$, not $bx$) It is also a convention that if $a^2 = e$(hence $a = a^{-1}$), then no

$x\;\;............\;\;xa$

for if $a = a^{-1}$, then multiplying by $a$ is the same as multiplying by $a^{-1}$

The Cayley diagram of a group contains the same information as the group's table. For instance, to find the product $(ab)(ab^2)$
in the previous figure 1, we start at $ab$ and follow the path corresponding to $ab^2$ (multiplying by $a$, then
by $b$, then again by $b$), which is (Figure 2)

Attachment 25766

This path leads to $b$, hence $(ab)(ab^2) = b$

As another example, the inverse of $ab^2$ is the path which leads from $ab^2$ back to $e$.
We note instantly that this is $ba$.

A point-and-arrow diagram is the Cayley diagram of a group iff it has the following two properties:

1) For each point $x$ and generator $a$, there is exactly one $a$-arrow ending at $x$;
furthermore, at most one arrow goes from $x$ to another point $y$.

2) If two different paths starting at $x$ lead to the same destination, then those two paths, starting at any point $y$, 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 $e$(neutral element), because
there is perfect symmetry in a Cayley diagram. Choose $e$, then label the diagram and proceed.)

Attachment 25763

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?
• Nov 17th 2012, 04:13 PM
chiro
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.
• Nov 17th 2012, 05:58 PM
x3bnm
Re: How to easily retrieve operation table of group elements from this Cayley diagram
Quote:

Originally Posted by chiro
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.

Thanks chiro for help. I solved the problem. Those who're interested read on:

Attachment 25770

where:
$b^3 = e$

$ab^2a = bab$

$b^2aba = bab^2$

$aba = b^2ab^2$

$a^2 = e$