So I have been reading a little about group theory and wanted to understand visually. I saw two different ways to visualize groups on many books and lectures.
$1$.Cayley graph:This is one way I found for visualizing groups . now , in a Cayley graph , there are some generators and relations .And by using that , I can make different kinds of groups.Here, they call each of the paths actions .This method is specially useful for visualizing things like Dihedral groups , orbits of a group (maybe homomorphism?) etc.
$2$.Group multiplication table : This one is another way to visualize the same group.It has a bunch of elements which combine with each other in the same manner as a multiplication table.This method is useful (for me at-least) to visualize things like cosets , lagranges theorem , normal subgroup , quotient group etc.
Now , one of the problems is that many books don't use both of the methods to describe the same group.So i get more biased for one method and can't understand anymore the concepts that were meant to be understood with the other method (or I am wrong again?)
In a abstract algebra textbook , I see two concepts :
$1$.Groups
$2$.Group Action
I saw the abstract definitions of groups and group actions (I will not state it here).Because Cayley diagram talk about actions ,I thought It could be related to group action somehow.And I already know that groups and group multiplication table fall hand in hand.
The problem is ,I know from the abstract definition that how is group actions came from groups from defintions .But I can't really see how cayley diagram(with generators and relations) and the abstract group action is related to each other. And if they are related which one I should use for intuitive studying? Also books juggle around with the definitions of group and group action , so I want to show the intuitive way to relate them with each other?(I mean mathematicians didn't come up with these definition out of thin air , right?)
Best Answer
Let $G $ be a group and $A $ a set. A (left) group action\begin{eqnarray} \star: G \times A &&\to A \\ (g,a) &&\mapsto g \star a \end{eqnarray} is a function from the cartesian product that is compatible with the groups structure ,i.e. $\forall a \in A, g,h \in G$ \begin{eqnarray} e\star a &&=a \\ g \star(h\star a) &&= (gh)\star a\end{eqnarray} where e is the neutral element.
The prime example would be the generic group action, a group acting on its own underlying set via leftmultiplication with a group element, that is A=G and \begin{eqnarray} \circ: G \times G &&\to G \\ (g,h) &&\mapsto g\circ h= gh \end{eqnarray} where we did nothing but relabel the groups inner operation for clarity. So this is obviously a group action because per definition $G$'s operation is associative and $e$ acts trivially on every element.
So how does this lead us to graphs?
A group action on a set A gives us an associated graph $X=(A,E)$ via $E=\{(g \star a,a): g \in G\}$. in case of the generic operation on the group $G$ on itself we optain a graphic representation of the group operation.
So this graph is really just the same thing as the multiplication table, for each element you can see what comes out if you multiply any element from the left.
We may notice that a lot of the edges in this graph are redundant. So we may specify and put special attention to a certain subset $ S \subset G$ and only add edges that represent leftmultiplication with elements from $(S \cup S^{-1}) \backslash \{e\}$, where $S^{-1}=\{s^{-1}: s \in S\}$ (omitting the neutral element since all it does is produce loops in the graph).
Usually one looks at so-called generating sets $S$ (that is there is no proper subgroup of $G$ containing $S$) and thats what i refer to as the Cayley-Graph of $G$ by the generator $S$
I interpret that as saying that the edges in the cayley graph represent the action of an element of the generating set. And although the "action" of S on G is not a group action in the forementioned sense (since $S$ is just a subset, not a group), it is however the restriction of the generic group action $\circ: G \times G \to G$ to $(S \cup S^{-1}) \backslash \{e\} \times G$,
There are quite a few theorems linking them. For example the cayley graph of a free group is a tree. What do you mean by relations in a Caleygraph?
Well use whatever works for you; for calculations tables seem more practical, for me graphs provide a better overview of the groups structure.