This is going to be a rather boring post. There are a few definitions that I need to get out of the way. After we get done with group actions I'm going to put group theory on hold for a while so that I can go through a crash course in linear algebra (and maybe start some topology) but later, when we revisit group theory, we're going to learn about representations and characters (which need linear algebra, hence the break) and at that point we'll need group actions and the definitions that I'm going to introduce here.
Kernel
We've already defined the kernel of a homomorphism and the kernel of a group action is very similar.
Definition:Kernel of an Action
Let $G$ be a group and $S$ a set and suppose $G$ acts on $S$. We call the kernel of the action the set of all elements in $G$ that fix every element of $S$, or $\left \{ g \in G : g \cdot \alpha = \alpha \; \forall \alpha \in S \right \}$.
This definition actually would have been unnecessary if we'd defined group actions in a different way. I didn't do it this way, but an equivalent definition of a group action of $G$ on $S$ would have been a homomorphism $\theta : G \to Perm \left( S \right )$ where $Perm \left( S \left)$ is the group of all permutations on the set, $S$. Then the kernel of the group action is equal to the kernel of $\theta$.
However, since we didn't define a group action as a homomorphism (in hindsight, maybe I should have but what do we do), the definition of a kernel needs some explanation. When a group element, $g \in G$, acts like the identity element on a particular set element, $s \in S$, in that $g \cdot s = s$ we say that $g$ fixes $s$. The kernel of the action is then the set of every group element that fixes every set element. In general, when we talk about the kernel of anything, what we mean is all the elements of a group that work like the identity element in a certain context, and this is the same for group actions.
Orbits and Stabilizers
The next two definitions introduce two important sets, the orbit and stabilizer of a set element. I'm going to spoil a surprise, but there's a cool theorem that we're going to learn soon called the Orbit Stabilizer Theorem which says that if $G$ acts on $S$ and $s \in S$ then $\left | G \right | = \left | \text{orb}_G \left ( s \right ) \right | \left | \text{stab}_G \left ( s \right ) \right |$.
Definition:Orbit
Let $G$ be a group and $S$ a set and suppose $G$ acts on $S$. Then if $s \in S$ we define the orbit of $s$ by $\text{orb}_G \left(s \right ) = \left \{ g \cdot s : g \in G \right \}$.
It might not be obvious from the definition, but $\text{orb}_G \left ( s \right )$ is actually a subset of $S$. Here's the way I think of it - if $s \in S$ then the orbit of $s$ is all of the elements of $S$ that you can get to using any element of $G$. More technically, as the definition says, the orbit is the set of all $g \cdot s$ where $g \in G$.
As it turns out, the orbits are a partition of $S$. It should be clear that $s \in \text{orb}_G \left ( s \right )$ since $e \cdot s = s$ so every orbit is non-empty. We also get that if $t \in \text{orb}_G \left ( s \right )$ then $\text{orb}_G \left ( s \right ) = \text{orb}_G \left ( t \right )$ (which is less clear but not so hard to prove). This gives us that each element of $S$ belongs to one and only one orbit, meaning that we have our partition.
The partition helps to explain the term "orbit." The orbits are isolated and if you start at $s$ and walk your way through $\text{orb}_G \left ( s \right )$ using the elements of $G$, you'll eventually get back to $s$.
Definition:Stabilizer
Let $G$ be a group and $S$ a set and suppose $G$ acts on $S$. Then if $s \in S$ we define the stabilizer of $s$ by $\text{stab}_G \left( s \right ) = \left \{ g \in G : g \cdot s = s \right \}$.
Orbits were subsets of $S$, but stabilizers are subsets of $G$. When we defined kernels, we defined them as all of the elements of $G$ that fix every single element of $S$ and stabilizers are similar in that the stabilizer of a particular element, $s \in S$ is the set of elements of $G$ that fix $s$.
Suppose $K$ is the kernel of the action. Based on the definitions, we get that $K \subseteq \text{stab}_G (s)$ for each $s \in S$. In fact, it is not hard to prove that
\[ K = \bigcap_{s \in S} \text{stab}_G (s) \]
and its also easy to prove that stabilizers are in fact subgroups of $G$.
Faithful and Transitive Actions
Shortly after we defined groups we defined all sorts of adjectives that we can use to describe groups like abelian, finite, and cyclic, and each of these adjective serve to add some extra axioms to our definition of a group. I'm now going to define some adjectives that we can use to add some extra axioms to the definition of group actions.
Definition:Faithful
Let $G$ be a group and $S$ a set and suppose $G$ acts on $S$. Then the action is called faithful if the kernel of the action is the trivial subgroup.
You can notice that the trivial action that we defined in the last entry is the opposite of faithful. The trivial action defines every element of the group to fix every element of the set so the kernel of the trivial action is the entire group. It turns out that the kernel of an action serves as a sort of measurement as to the triviality of the action which is meaningful since we learned that every group acts on every set. A small kernel means that the action is interesting and a large kernel gives us that the action is less inventive. When an action is faithful we know that we're dealing with a "good" action.
Definition:Transitive
Let $G$ be a group and $S$ a set and suppose $G$ acts on $S$. Then the action is called transitive if it has only one orbit.
We noted earlier that orbits form a partition of $S$. In transitive actions, this the orbits form the most boring possible partition in that there is only one orbit and $\text{orb}_G (s) = \text{orb}_G (t)$ for every $s,t \in S$. This may seem kind of unfortunate, but actually its very cool and useful. If an action is transitive then for any $s$ and $t$ in $S$ there is some $g \in G$ such that $g \cdot s = t$, which is useful from time to time.
References
Previous Related Post: Group Action Examples
Text Reference: Gallain Chapter 7
Wikipedia: Group Action
Wolfram Mathworld: Group Orbit
Wolfram Mathworld: Stabilizer
Wolfram Mathworld: Faithful Group Action
Wolfram Mathworld: Transitive Group Action
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