Group Action Attributes

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

Homomorphisms

Group homomorphisms are very closely related to group isomorphisms. In fact, it turns out that every isomorphism is a homomorphism and as such, homomorphisms can be viewed as a generalization of isomorphisms. These very important functions are fundamental in the world of algebra and are one of the most important tools that we have. As we'll see in this and in upcoming posts, the application of homomorphisms to various groups gives a lot of useful results.

Definition: Group Homomorphism
A function, φ, from a group G to a group G* is a group homomorphism if it preserves the group operation - that is if ∀a,b∈G, φ(ab) = φ(a)φ(b).

A homomorphism is a function that has the operation preserving property that we described for isomorphisms without regard to bijectiveness. Of course, homomorphisms can be bijective, explaining why each isomorphism is a homomorphism. This means that it is easier for a function to be a homomorphism than an isomorphism, but it also means that homomorphisms have more interesting properties. Isomorphisms are used to identify groups as isomorphic, but not for much else, whereas homomorphisms provide us with a lot of usefullness, mostly as a result of the following sets.

Definition: Kernel
The kernel of a homomorphism, φ:G→G*, is the set {g∈G : φ(g)=e} and it is denoted by Ker(φ) or Kerφ.

Definition: Image
The image of a homomorphism, φ:G→G*, is the set {φ(x) : x∈G} and is denoted by Im(φ) or Imφ.

The first thing to notice here is that if φ:G→G* is a homomorphism, then Kerφ⊆G and Imφ⊆G*. The kernel is all of the things in G that are mapped to the identity, and the image is all of the things in G* that can be attained by a mapping of φ. It is important to note that these sets are trivial in the case that φ is actually an isomorphism - that is that in this case, Kerφ={e} and Imφ=G*. This does give another convenient way to check if a function is an isomorphism. Instead of proving that φ is bijective, it suffices to show that Kerφ={e} and Imφ=G* (and, of course, that φ is operation preserving). However, we get much more than that. It turns out that both the image and the kernel are subgroups.

Theorem: Kernels are Subgroups
Let φ:G→G* be a group homomorphism. Then Kerφ is a subgroup of G.

Proof:

Let φ:G→G* be a group homomorphism and choose a,b∈Kerφ. We then have that φ(ab^-1) = φ(a)φ(b)^-1 = e(e^-1) = e and thus ab^-1∈Kerφ. Therefore, by the one-step subgroup test, Kerφ is a subgroup of G.

Theorem: Images are Subgroups
Let φ:G→G* be a group homomorphism. Then Imφ is a subgroup of G*.

Proof:

Let φ:G→G* be a group homomorphism and choose x,y∈Imφ. Then ∃a,b∈G such that φ(a)=x and φ(b)=y. Now, φ(ab^-1) = φ(a)φ(b)^-1 = xy^-1 and thus xy^-1∈Imφ. Therefore, by the one-step subgroup test, Imφ is a subgroup of G*.

In those proofs I used a fact that I have not yet proved - specifically that φ(b^-1 = φ(b)^-1. This is true, though, and is an easy thing to prove. The fact that these two sets are subgroups is important and very convenient, but we get even more than that.

Theorem: Kernels are Normal
Let φ:G→G* be a group homomorphism. Then Kerφ⊳G.

Proof:

Let φ:G→G* be a group homomorphism and choose g∈G and n∈Kerφ. Then φ(gng^-1) = φ(g)φ(n)φ(g)^-1 = φ(g)eφ(g)^-1 = φ(g)φ(g)^-1 = e thus giving that gng^-1∈Kerφ. Since the choice of n∈Kerφ was arbitrary, gKerφg^-1⊆Kerφ for g∈G. Thus, by the normal subgroup test, Kerφ⊳G

It turns out that images are not normal subgroups. This is not all that surprising when you really think about what an image is in relation to a codomain. However, the fact that kernels are normal subgroups is a surprisingly wonderful fact and we get a lot of mileage out of it. In the next post, we'll see a very useful result called the first group isomorphism theorem that uses the normality of kernels in a very natural way.

References

Previous Related Post: Properties of Isomorphisms

Text Reference: Gallain Chapter 10

Wikipedia: Group Homomorphism

Wolfram Mathworld: Group Homomorphism

Planet Math: Group Homomorphism