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

1. ## 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 $\displaystyle G$ has only one generator $\displaystyle a$ (that is, $\displaystyle G$ is the cyclic group $\displaystyle \langle a \rangle$), then the arrow $\displaystyle \rightarrow$
represents the operation "multiply by $\displaystyle a$":

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

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

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

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

$\displaystyle x \rightarrow xb$

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

$\displaystyle x \leftarrow xb^{-1}$

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

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

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

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

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

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

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

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

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

2. ## 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.

3. ## Re: How to easily retrieve operation table of group elements from this Cayley diagram

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:

where:
$\displaystyle b^3 = e$

$\displaystyle ab^2a = bab$

$\displaystyle b^2aba = bab^2$

$\displaystyle aba = b^2ab^2$

$\displaystyle a^2 = e$

### every finite group may be represented by a diagram known as a Cayley

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