I am going over the group action and unfortunately I am not sure if I understand it very well. So for this reason, I have a few questions that possibly could help me to understand this concept better. A group action is defined as: Group $G$ acts on a set $X$ if there is a homomorphism $\sigma:G \rightarrow S_X$. I am aware group action has other definitions but for me this one is easier to understand.
Now $S_X$ is the group of permutations of $X$. It is not crystal clear what is meant by this to me. Is $S_X$ the same thing as $S_{|X|}$? When we talk about $S_n$, I have a clear mental picture; $n$ is an integer and I think of all the permutations with $n$ elements. But in $S_X$, $X$ is not an integer but a set. So suppose $X=\{1,2,3\}$. Is $S_X$ the same thing as $S_3$ in this case?
The other part of my question is about what the definition implies: it says "… if there is a homomorphism". Is it possible that homomorphism does not exist? Is the homomorphism unique? Or is it possible to have more than one homomorphism? Once we specify a group and a set, how do we find such homomorphism(s)? Is there a requirement for the size of the group and the set to make this definition work, for instance $|G| \geq |X|$?
Please let me know if you would like me to clarify my question. Any help with making a better intuition is appreciated!
Best Answer
For your first question, $S_X$ is defined as the group of bijections from $X$ to $X$. The group operation is function composition. If $X$ is finite, say with $n$ elements, then the groups $S_X$ and $S_n$ are obviously (noncanonically) isomorphic.
There is always at least one group homomorphism from $G$ to $S_X$, namely the one which sends everything in $G$ to the identity map in $S_X$. This is the "stupid group action" which doesn't do anything ($g.x = x$ for all $g \in G$ and $x \in X$).
There are typically many homomorphisms from $G$ to $S_X$. Therefore, there are typically many group actions of $G$ on $X$.
"Once we specify a group and a set, how do we find such homomorphisms?"
Good question. In many settings, the group action comes up naturally. It's not like people are taking random sets $X$ and $G$ and asking, "I wonder how many different actions of $G$ I can find on $X$." I mean maybe some people are doing this, but usually the group action is a convenient way to describe some existing phenomenon they are trying to study.
Example: Let $X = \{ z \in \mathbb C: \operatorname{Im}(z) > 0\}$ be the upper half plane, and let $G = \operatorname{SL}_2(\mathbb Z)$ be the group of integer matrices with determinant $\pm 1$. There is a natural group action of $G$ on $X$ by
$$\begin{pmatrix} a & b \\ c & d \end{pmatrix}.z = \frac{az+b}{cz+d}.$$
This came up (I believe) when people were working out problems like describing all holomorphic bijections from the Riemann sphere to itself, and it turns out that they all look like the above (where the matrix can be anything in $\operatorname{GL}_2(\mathbb R)$).
One example where people are taking certain groups $G$ and certain sets $X$, and asking what are the ways $G$ can act on $X$, is when $X$ is a vector space. The bijections from $X$ to itself coming from the elements of $G$ are required to also be linear transformations on $X$. Such group actions are called representations. Representation theory includes the problem of describing all representations of a given group on a given vector space, and this turns out to be an extremely difficult problem in general. Finding such group actions is no simple matter.