
# 6.1: Orbits and Stabilizers

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In this section, we'll examine orbits and stabilizers, which will allow us to relate group actions to our previous study of cosets and quotients.

Definition 6.1.0: The Orbit

Let $$S$$ be a $$G$$-set, and $$s\in S$$. The orbit of $$s$$ is the set $$G\cdot s = \{g\cdot s \mid g\in G\}$$, the full set of objects that $$s$$ is sent to under the action of $$G$$.

There are a few questions that come up when encountering a new group action. The foremost is 'Given two elements $$s$$ and $$t$$ from the set $$S$$, is there a group element such that $$g\cdot s=t$$?' In other words, can I use the group to get from any element of the set to any other? In the case of the action of $$S_n$$ on a coin, the answer is yes. But in the case of $$S_4$$ acting on the deck of cards, the answer is no. In fact, this is just a question about orbits. If there is only one orbit, then I can always find a group element to move from any object to any other object. This case has a special name.

Definition 6.1.1: Transitive Group Action

A group action is transitive if $$G\cdot s = S$$. In other words, for any $$s, t\in S$$, there exists $$g\in G$$ such that $$g\cdot s=t$$. Equivalently, $$S$$ contains a single orbit.

Equally important is the stabilizer of an element, the subset of $$G$$ which leaves a given element $$s$$ alone.

Definition 6.1.2: The Stabilizer

The stabilizer of $$s$$ is the set $$G_s = \{g\in G \mid g\cdot s=s \}$$, the set of elements of $$G$$ which leave $$s$$ unchanged under the action.

For example, the stabilizer of the coin with heads (or tails) up is $$A_n$$, the set of permutations with positive sign. In our example with $$S_4$$ acting on the small deck of eight cards, consider the card $$4D$$. The stabilizer of $$4D$$ is the set of permutations $$\sigma$$ with $$\sigma(4)=4$$; there are six such permutations.

In both of these examples, the stabilizer was a subgroup; this is a general fact!

Proposition 6.1.3

The stabilizer $$G_s$$ of any element $$s \in S$$ is a subgroup of $$G$$.

Proof 6.1.4

Let $$g, h \in G_s$$. Then $$gh\cdot s = g\cdot (h\cdot s) = g\cdot s=s$$. Thus, $$gh\in G_s$$. If $$g\in G_s$$, then so is $$g^{-1}$$: By definition of a group action, $$1\in G_s$$, so:
$$s=1\cdot s = g^{-1}g\cdot s = g^{-1}s$$.

Thus, $$G_s$$ is a subgroup.

#### Group action morphisms

And now some algebraic examples!

1. Let $$G$$ be any group and $$S=G$$. The left regular action of $$G$$ on itself is given by left multiplication: $$g\cdot h = gh$$. The first condition for a group action holds by associativity of the group, and the second condition follows from the definition of the identity element. (There is also a right regular action, where $$g\cdot h = hg$$; the action is 'on the right'.) The Cayley graph of the left regular action is the same as the usual Cayley graph of the group!

2. Let $$H$$ be a subgroup of $$G$$, and let $$S$$ be the set of cosets $$G/\mathord H$$. The coset action is given by $$g\cdot (xH) = (gx)H$$.

The coset action is quite special; we can use it to get a general idea of how group actions are put together.
Proposition 6.1.6

Let $$S$$ be a $$G$$-set, with $$s\in S$$ and $$G_s$$. For any $$g, h\in G$$, $$g\cdot s=h\cdot s$$ if and only if $$gG_s=hG_s$$. As a result, there is a bijection between elements of the orbit of $$s$$ and cosets of the stabilizer $$G_s$$.

Proof 6.1.7

We have $$gG_s=hG_s$$ if and only if $$h^{-1}g\in G_s$$, if and only if $$(h^{-1}g)\cdot s=s$$, if and only if $$h\cdot s=g\cdot s$$, as desired.

In fact, we can generalize this idea considerably. We're actually identifying elements of the $$G$$-set with cosets of the stabilizer group, which is also a $$G$$-set; in other words, defining a function $$\phi$$ between two $$G$$-sets. The theorem says that this function preserves the group operation: $$\phi(g\cdot s)=g\cdot \phi(s)$$.

Definition

Let $$S, T$$ be $$G$$-sets. A morphism of $$G$$-sets is a function $$\phi:S\rightarrow T$$ such that $$\phi(g\cdot s)=g\cdot \phi(s)$$ for all $$g\in G, s\in S$$. We say the $$G$$-sets are isomorphic if $$\phi$$ is a bijection.

We can then restate the proposition:

Theorem 6.1.9

For any $$s$$ in a $$G$$-set $$S$$, the orbit of $$S$$ is isomorphic to the coset action on $$G_s$$.

Now we can use LaGrange's theorem in a very interesting way! We know that the cardinality of a subgroup divides the order of the group, and that the number of cosets of a subgroup $$H$$ is equal to $$|G|/\mathord |H|$$. Then we can use the relationship between cosets and orbits to observe the following:

Theorem 6.1.10

Let $$S$$ be a $$G$$-set, with $$s\in S$$. Then the size of the orbit of $$s$$ is $$|G|/\mathord |G_s|$$.

For a somewhat obvious example, considering $$S_{13}$$ acting on the numerical values of playing cards, we can observe that any given card is fixed by a subgroup of $$S_{13}$$ isomorphic to $$S_{12}$$ (switching around the other twelve numbers in any way doesn't change affect the given card). Then the size of the orbit of the card is $$|S_{13}|/\mathord |S_{12}| = 13$$. That's a number we could have figured out directly by reasoning a bit, but it shows us that the theorem is working sensibly!

Now that we have a notion of isomorphism of $$G$$-sets, we can say something to classify $$G$$-sets. What kinds of actions are possible?

Let $$G$$ be a finite group, and $$S$$ a finite $$G$$-set. Then $$S$$ is a collection of orbits. We knw that every orbit is isomorphic to $$G$$ acting on the cosets of some subgroup of $$H$$. So we have the following theorem:

Theorem 6.1.11: Classification of $$G$$-Sets
Let $$G$$ be a finite group, and $$S$$ a finite $$G$$-set. Then $$S$$ is isomorphic to a union of coset actions of $$G$$ on subgroups.

For example, $$S_{13}$$ acting on a full deck of cards decomposes as a union of four orbits, each isomorphic to the coset action of $$S_{13}$$ on a subgroup isomorphic to $$S_{12}$$.

In short, to understand all possible $$G$$-sets, we should try to understand all of the subgroups of $$G$$. In general, this is a hard problem, though it's easy for some cases.

Exercise 6.1.12

1. For $$n=15$$, draw Cayley graphs of the coset action of $$\mathbb{Z}_{15}$$ on each of it's cosets.
2. Describe all the subgroups of $$\mathbb{Z}_n$$ for arbitrary $$n$$.

Exercise 6.1.13

$$S_n$$ acts on subsets of $$N=\{1,2,3,\ldots,n\}$$ in a natural way: if $$U=\{i_1, \ldots, i_k\}\subset N$$, then $$\sigma\cdot U = \{\sigma(i_1), \ldots, \sigma(i_k)\}$$.

1. Decompose the action of $$S_4$$ on the subsets of $$\{1,2,3,4\}$$ into orbits.
2. Draw a Cayley graph of the action.
3. Identify each orbit with the coset action on a subgroup of $$S_4$$.

### Contributors

• Tom Denton (Fields Institute/York University in Toronto)