6.2: Orbits and Stabilizers

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 \(g\cdot (xH) = (gx)H\).

Figure 6.1: \(H\) is the subgroup of \(S_4\) with \(\sigma(1)=1\) for all \(\sigma\) in \(H\). This illustrates the action of \(S_4\) on cosets of \(H\).

Consider the permutation group \(S_n\), and fix a number \(i\) such that \(1\leq i\leq n\). Let \(H_i\) be the set of permutations in \(S_n\) with \(\sigma(i)=i\).

1. Show \(H_i\) is a subgroup of \(S_n\).
2. Now let \(n=5\) and Sketch the Cayley graph of the coset action of \(S_5\) on \(H_1\) and \(H_3\).

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 \(n\).

\(S_n\) 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 and Attributions

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