Group actions

A group action is a representation of the elements of a group as symmetries of a set. Many groups have a natural group action coming from their construction; e.g. the dihedral group \(D_4\) acts on the vertices of a square because the group is given as a set of symmetries of the square. A group action of a group on a set is an abstract generalization of this idea, which can be used to derive useful facts about both the group and the set it acts on.

Formally, a group action of a group \(G\) on a set \(X\) is a function \(f \colon G \times X \to X\) satisfying the following properties:

(1) \(f(e_G,x) = x\) for all \(x\in X\)

(2) \(f(gh,x) = f\big(g,f(h,x)\big)\) for all \(g,h \in G\) and \(x\in X.\)

When the action is clear, the function \(f(g,x)\) is often written as \(g \cdot x.\) With this notation, the axioms become

(1) \(e_G \cdot x = x\)

(2) \( g \cdot (h \cdot x) = (gh) \cdot x\).

The standard example of a group action is when \(G\) equals the symmetric group \(S_n\) \((\)or a subgroup of \(S_n)\) and \(X = \{1,2,\ldots,n\}\). Then \(G\) acts on \(X\) by the formula \(g \cdot x = g(x).\) The properties are clear: \(e \cdot x = e(x) = x\) when \(e\) is the identity of \(S_n,\) and \(g \cdot (h \cdot x) = g \cdot h(x) = g\big(h(x)\big) = (g \circ h)(x).\)

Here are several basic concepts related to group actions.

Let \(G\) be a group acting on a set \(X.\)

A fixed point of an element \(g \in G\) is an element \(x \in X\) such that \(g \cdot x = x.\)
The stabilizer \(G_x\) of a point \(x \in X\) is the set of elements \(g \in G\) such that \(x\) is a fixed point of \(g.\)
The orbit of an element \(x \in X\) is the set of elements \(y \in X\) such that \(g \cdot x = y\) for some \(g \in G.\)

Let \(G = {\mathbb Z}_2 = \{e,g\} \) and \(X = {\mathbb Z}.\)
Let \(G\) act on \(X\) by the formulas \(e\cdot x = x\) and \(g\cdot x = -x.\)

Find the fixed points of every element of \(G.\)
Find the stabilizer of every element of \(X.\)
Find the orbits of the elements of \(X.\)

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Every element of \(X\) is a fixed point of \(e,\) but \(g\) has exactly one fixed point, namely \(0.\)

The stabilizer of \( 0\) is all of \(G,\) but the stabilizer of any other element of \(X\) is the trivial subgroup \( \{e\} \) of \(G.\)

The orbit of \( 0\) is \( \{0\},\) and the orbit of any other element \(x \) is a two-element set \( \{-x,x\}.\) \(_\square\)

Note that \(X\) is partitioned as a union of its distinct and disjoint orbits. This is true in general; it follows from the fact that the relation defined by \( x\sim y\) if \(y\) is in the orbit of \(x\) is an equivalence relation.

Certain common properties related to these definitions arise in many group actions.

Let \(G\) be a group acting on a set \(X.\)

The action is transitive if there is only one orbit: for any \(x,y \in X,\) there is an element \(g \in G\) such that \( g\cdot x = y.\)
The action is faithful if the intersection of the stabilizers \(G_x\) for \(x \in X\) consists only of the trivial element \( e_G.\)

The axioms of a group action give a group homomorphism \( G \to \text{Sym}(X),\) where \( \text{Sym}(X)\) is the group of permutations of the elements of \(X.\) A faithful action, then, is one for which this homomorphism is injective, as its kernel is trivial.

There are many examples where a group \(G\) acts on objects related to \(G.\) Here are some examples:

(1) Every group acts on itself by left multiplication: \(G\) acts on \(G\) via the formula \(g \cdot x = gx.\) This is a transitive and faithful action; there is one orbit, and in fact the stabilizer of any element \(x\) is trivial: \(gx=x\) if and only if \(g\) is the identity.

(2) Every group acts on itself by conjugation: \(G\) acts on \(G\) via the formula \(g \cdot x = gxg^{-1}.\) The orbits of this action are called conjugacy classes, and the stabilizer of an element \(x\) is called the centralizer \(C_G(x).\)

(3) If \(H\) is a subgroup of \(G,\) then \(G\) acts on the set of cosets \(G/H\) by left multiplication. The action is transitive, since \(\big(kg^{-1}\big)(gH) = kH\) for any \(g,k \in G.\) This induces a map \( G \to \text{Sym}(G/H)\) whose kernel is equal to the intersection of all the conjugates of \(H,\) which is the largest normal subgroup contained in \(H.\) So if \(H\) does not contain any nontrivial normal subgroups, the action is faithful. This construction is quite useful in the analysis of simple groups.

Let \({\mathbb P}^1_{\mathbb C}\) be the projective complex line: this can be thought of as the set of complex numbers plus the point \(\infty.\) Then the group \( GL_2({\mathbb C}) \) of invertible \(2\times 2\) matrices with complex entries acts on \( {\mathbb P}^1\) via the formula

\[\begin{pmatrix} a&b \\ c&d \end{pmatrix} \cdot z = \frac{az+b}{cz+d},\]

with the understanding that \( -d/c\) maps to \(\infty\) and \(\infty\) maps to \(a/c.\) \((\)If \(c=0,\) then \(\infty\) maps to \(\infty.)\)

Checking that this is a group action boils down to the following computation for the second axiom:

\[\begin{align} \begin{pmatrix} a&b\\c&d \end{pmatrix} \cdot \left( \begin{pmatrix} e&f\\g&h \end{pmatrix} \cdot z \right)&= \begin{pmatrix} a&b\\c&d \end{pmatrix} \cdot \frac{ez+f}{gz+h} \\ &= \frac{a\left(\frac{ez+f}{gz+h}\right) +b}{c\left(\frac{ez+f}{gz+h}\right)+d} \\ &= \frac{a(ez+f)+b(gz+h)}{c(ez+f)+d(gz+h)} \\ &= \frac{(ae+bg)z+(af+bh)}{(ce+dg)z+(cf+dh)} \\ &= \begin{pmatrix} ae+bg & af+bh \\ ce+dg & cf+dh \end{pmatrix} \cdot z \\ \end{align}\]

and the matrix \( \begin{pmatrix} ae+bg & af+bh \\ ce+dg & cf+dh \end{pmatrix} \) is the product \( \begin{pmatrix} a&b\\c&d \end{pmatrix} \begin{pmatrix} e&f\\g&h \end{pmatrix},\) which shows that this gives a group action.

Note that the matrix \( \begin{pmatrix} a&0 \\ 0&a \end{pmatrix} \) acts trivially on the projective line, since \(\frac{az}{a} = z.\) So the action of \( GL_2({\mathbb C}) \) turns into an action of \(GL_2({\mathbb C})/N,\) where \(N\) is the subgroup of matrices which are constant multiples of the identity. This group is called the projective general linear group \(PGL_2({\mathbb C});\) the action of \(PGL_2({\mathbb C})\) is faithful and also transitive. (In fact, it is 3-transitive: any list of three distinct points can be sent to any other list of three distinct points.)

There is a natural relationship between orbits and stabilizers of a group action.

Let \(G\) be a group acting on a set \(X.\) Fix a point \(x\in X\) and consider the function \(f_x \colon G \to X\) given by \(g \mapsto g \cdot x.\) In fact, this gives a function \( h_x \colon G/G_x \to X\) given by the same formula, where \(G_x\) is the stabilizer of \(x.\) Here \(G/G_x\) is the set of left cosets of \(G_x.\) Note that \(G_x \) is not necessarily a normal subgroup. So \(G/G_x\) is just a set of cosets, not a quotient group.

The function \(h_x\) is injective, and its image is precisely the orbit of \(x.\) The conclusion is as follows:

There is a bijection between the set \( G/G_x\) and the orbit of \(x.\)

When the orbit of \(x\) is finite, this gives the following formula:

Let \(G\) be a group acting on a set \(X.\) Let \(G_x\) be the stabilizer of an element \(x \in X.\) Suppose that the orbit \( O_x\) of \(x\) is finite. Then the index \([G:G_x]\) is finite and equal to \( |O_x|.\) If \(G\) is finite, then

\[|G_x| \cdot |O_x| = |G|.\]

This theorem is used to prove many useful facts in group theory, including Burnside's lemma and the class formula. Here are two explicit examples of the theorem in action.

Let \(G = S_n\) and let \(X = \{ 1,\ldots,n\}.\) Let \(x = n.\) The orbit of \(x\) is all of \(X\) (the action is transitive), so the theorem says that the index \([G:G_x]\) is equal to \(n.\) Indeed, \(G_x\) is isomorphic to \(S_{n-1},\) the group of permutations of \( \{ 1, \ldots, n-1 \}.\) So \(|G|=n!,\) \(|G_x| = (n-1)!,\) and \([G:G_x] = |G|/|G_x| = n!/(n-1)! = n.\)

How many rotational symmetries does a cube have? (Extra credit: what is the group of symmetries of the cube?)

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The question is asking for the number of ways to permute the vertices (or faces, or edges) of a cube by twisting and turning (but not reflecting) the cube in 3D-space. The orbit-stabilizer theorem can be used to solve this problem in three different ways.

Let \(G\) be the group of symmetries of the cube. Then \(G\) acts on the set \(F\) of faces of the cube. There are six faces, and the action is transitive, so the size of an orbit \(O_f\) of a given face is \(6.\) The order of the stabilizer \( G_f\) of a face is 4, because there are four rotations of a cube that fix a face (the rotations around the axis perpendicular to the face). So by the orbit-stabilizer theorem, \(|G| = 4 \cdot 6 = 24.\)

But \(G\) also acts on the vertices of a cube. There are eight vertices, and again the action is transitive. What is the order of the stabilizer \(G_x\) of a vertex? The only rotations of a cube that leave one vertex fixed are the three rotations of \(120,240,\) and \(360\) degrees around the body diagonal of the cube. So \(|G_x|=3\) and \(|O_x|=8,\) so by the orbit-stabilizer theorem \( |G| = 3\cdot 8 = 24.\)

And \(G\) also acts on the edges of a cube. There are twelve edges, and the action is transitive. The stabilizer \(G_e\) of an edge has order two; there is the identity of \(G,\) and the unique element of \(G\) which switches the vertices of the edge. So again \(|G| = 2 \cdot 12 = 24.\) \(_\square\)

In fact, \(G\) is isomorphic to the symmetry group \(S_4,\) and the isomorphism uses yet another action of \(G,\) namely the action of \(G\) on the body diagonals, of which there are four. This action gives a homomorphism \(G \to S_4,\) and it is straightforward to show that this homomorphism is injective (the action is faithful), and the groups have the same size, so it must be an isomorphism.

• Burnside's Lemma

• Conjugacy Classes