Introduction to Groups and Geometries

Section2.5Group actions

Definition2.5.1.Group action, orbit, stabilizer.

Let $$G$$ be a group and let $$X$$ be a set. An action of the group $$G$$ on the set $$X$$ is a group homomorphism
\begin{equation*} \phi \colon G\to \Perm(X). \end{equation*}
We say that the group $$G$$ acts on the set $$X\text{,}$$ and we call $$X$$ a $$G$$-space. For $$g\in G$$ and $$x\in X\text{,}$$ we write $$gx$$ to denote $$(\phi(g))(x)\text{.}$$
1
Other notations for $$(\phi(g))(x)$$ are $$g(x)\text{,}$$ $$g\cdot x\text{,}$$ and $$g.x\text{.}$$
We write $$\Orb(x)$$ to denote the set
\begin{equation*} \Orb(x)=\{gx\colon g\in G\}, \end{equation*}
called the orbit of $$x\text{,}$$ and we write $$\Stab(x)$$ to denote the set
\begin{equation*} \Stab(x) = \{g\in G\colon gx=x\}, \end{equation*}
called the stabilizer or isotropy subgroup
2
It must be proved that $$\Stab(x)$$ is indeed a subgroup of $$G\text{.}$$ See Checkpoint 2.5.3 below.
of $$x\text{.}$$ A group action is transitive if there is only one orbit. A group action is faithful if the map $$G\to \Perm(X)$$ has a trivial kernel.

Checkpoint2.5.2.

Find the indicated orbits and stabilizers for each of the following group actions.
1. $$D_4$$ acts on the square $$X=\{(x,y)\in \R^2\colon -1\leq x,y\leq 1\}$$ by rotations and reflections. What is the orbit of $$(1,1)\text{?}$$ What is the orbit of $$(1,0)\text{?}$$ What is the stabilizer of $$(1,1)\text{?}$$ What is the stabilizer of $$(1,0)\text{?}$$
2. The circle group $$S^1$$ (see Subsection 2.1.3) acts on the two-sphere $$S^2$$ by rotation about the $$z$$-axis. Given an element $$e^{i\alpha}$$ in $$S^1$$ a point $$(\theta,\phi)$$ in $$S^2$$ (in spherical coordinates), the action is given by
\begin{equation*} e^{i\alpha}\cdot (\theta,\phi)=(\theta,\phi+\alpha). \end{equation*}
What is the orbit of $$(\pi/4,\pi/6)\text{?}$$ What is the orbit of the north pole $$(0,0)\text{?}$$ What is the stabilizer of $$(\pi/4,\pi/6)\text{?}$$ What is the stabilizer of the north pole?
3. Any group $$G$$ acts on itself by conjugation, that is, by $$(\phi(g))(x)=gxg^{-1}=C_g(x)$$ (see Exercise 2.4.10). Describe the orbit and stabilizer of a group element $$x\text{.}$$
1. $$\Orb((1,1))=\{(1,1),(1,-1),(-1,1),(-1,-1)\}\text{,}$$ $$\Orb((1,0))=\{(1,0),(-1,0),(0,1),(0,-1)\}\text{,}$$ $$\Stab((1,1))= \{R_0,F_{D'}\}\text{,}$$ $$\Stab((1,0))=\{R_0,F_H\}$$
2. $$\Orb(\pi/4,\pi/6)$$ is the horizontal circle on $$S^2$$ with "latitude" $$\pi/4\text{,}$$ $$\Orb(0,0)=\{(0,0)\}\text{,}$$ $$\Stab(\pi/4,\pi/6)=\{1\}\text{,}$$ $$\Stab{(0,0)}=S^1$$
3. $$\Orb(x)=\{gxg^{-1}\colon g\in G\}\text{,}$$ $$\Stab(x)=C(x)$$ (the centralizer of $$x$$)

Checkpoint2.5.3.

Show that the stabilizer of an element $$x$$ in a $$G$$-space $$X$$ is a subgroup of $$G\text{.}$$

Definition2.5.4.Orbit space.

We write $$X/G$$ to denote the set
\begin{equation*} X/G=\{\Orb(x)\colon x\in X\} \end{equation*}
of orbits of the group $$G$$ acting on a set $$X\text{.}$$ The set $$X/G$$ is also called the orbit space of the group action.

ExercisesExercises

1.The sign of a permutation.

In this exercise we establish a function $$\sgn\colon S_n\to \{-1,+1\}\text{,}$$ called the sign function. We start by defining an action of the symmetric group $$S_n$$ on a set of two polynomials. Let $$\Delta$$ be the polynomial
\begin{equation*} \Delta = \prod_{1\leq i\lt j\leq n} (x_i-x_j) \end{equation*}
in the variables $$x_1,x_2,\ldots,x_n\text{.}$$ Let $$X$$ be the set $$X=\{\Delta,-\Delta\}\text{.}$$ Given a permutation $$\alpha$$ in $$S_n\text{,}$$ let $$\alpha \Delta$$ be the polynomial
\begin{equation*} \alpha \Delta = \prod_{1\leq i\lt j\leq n} (x_{\alpha(i)}-x_{\alpha(j)}) \end{equation*}
and let $$\alpha(-\Delta) = -\alpha \Delta\text{.}$$ In the exercises below, you will show that $$\alpha\Delta=\pm \Delta$$ for all $$\alpha$$ in $$S_n\text{.}$$ This allows us to define the sign of a permutation $$\alpha$$ to be $$+1$$ or $$-1$$ according to whether $$\alpha\Delta=\Delta$$ or $$\alpha\Delta=-\Delta\text{,}$$ respectively.
\begin{equation*} \sgn(\alpha) = \begin{cases} +1 \amp \text{if }\amp \alpha\Delta = \Delta\\ -1 \amp \text{if } \amp \alpha\Delta = -\Delta \end{cases} \end{equation*}
We say that a permutation $$\alpha$$ is even if $$\sgn(\alpha)=+1\text{,}$$ and we say $$\alpha$$ is odd if $$\sgn(\alpha)=-1\text{.}$$
1. Write out $$\Delta$$ for $$n=4$$ variables without using the product notation symbol $$\prod\text{.}$$
2. Continuing with $$n=4\text{,}$$ write out $$(134)\Delta$$ and $$(1324)\Delta\text{.}$$ In both cases, match the factors with $$\Delta$$ and identify which factors experience a sign change. Use this calculation to find $$\sgn(134)$$ and $$\sgn(1324)$$ directly and explicitly from the definition.
3. Show that $$\alpha \Delta = \pm \Delta$$ for all $$\alpha$$ in $$S_n\text{.}$$
4. Show that $$\tau\Delta=-\Delta$$ for any transposition $$\tau$$ in $$S_n\text{.}$$
5. Show that $$\phi\colon S_n \to \Perm(X)$$ given by $$\phi(\alpha)(\pm \Delta) = \alpha (\pm \Delta)$$ is a group action.
6. Use the group action property $$(\alpha\beta)\Delta = \alpha(\beta\Delta)$$ to explain why $$\sgn(\alpha\beta)=\sgn(\alpha)\sgn(\beta)$$ for all $$\alpha,\beta$$ in $$S_n\text{.}$$ In other words, the sign function is a homomorphism of groups $$S_n\to C_2\text{.}$$
7. Conclude that, if $$\alpha$$ is expressible as a product of an even number of transpositions, then $$\alpha$$ is an even permutation, and that any product of transpositions that equals $$\alpha$$ must have an even number of transpositions. (A similar statement holds replacing the word "even" by the word "odd".)
8. The subset of even permutations of $$S_n$$ is denoted $$A_n\text{.}$$ Give two arguments why $$A_n$$ is a normal subgroup of $$S_n\text{.}$$ Use (i) the 1-step or the 2-step subgroup test and definition Definition 2.4.8, and (ii) using criterion 1 of Proposition 2.4.9. This group $$A_n$$ is called the alternating group.

2.Actions of a group on itself.

Let $$G$$ be a group. Here are three actions $$G\to \Perm(G)$$ of $$G$$ on itself. Left multiplication is given by
\begin{equation*} g\to L_g \end{equation*}
where $$L_g$$ is given by $$L_g(h)=gh\text{.}$$ Right inverse multiplication is given by
\begin{equation*} g\to R_g \end{equation*}
where $$R_g$$ is given by $$R_g(h)=hg^{-1}\text{.}$$ Conjugation is given by
\begin{equation*} g\to C_g \end{equation*}
where $$C_g$$ is given by $$C_g(h)=ghg^{-1}\text{.}$$
1. Show that, for $$g\in G\text{,}$$ the maps $$L_g,R_g,C_g$$ are elements of $$\Perm(G)\text{.}$$
2. Show that each of these maps $$L,R,C$$ is indeed a group action.
3. Show that the map $$L$$ is injective, so that $$G\approx L(G)\text{.}$$
Consequence of this exercise: Every group is isomorphic to a subgroup of a permutation group.

3.Cosets, revisited.

Let $$H$$ be a subgroup of a group $$G\text{,}$$ and consider the map
\begin{equation*} R\colon H\to \Perm(G) \end{equation*}
given by $$h\to R_h\text{,}$$ where $$R_h(g)=gh^{-1}$$ (this is the restriction of right inverse multiplication action in Exercise 2.5.2 to $$H$$). Show that the orbits of this action of $$H$$ on $$G$$ are the same as the cosets of $$H\text{.}$$ This shows that the two potentially different meanings of $$G/H$$ (one is the set of cosets, the other is the set of orbits of the action of $$H$$ on $$G$$ via $$R$$), are in fact in agreement.

4.The natural action of a matrix group on a vector space.

Let $$G$$ be a group whose elements are $$n\times n$$ matrices with entries in a field $$\F$$ and with the group operation of matrix multiplication. The natural action $$G\to \Perm(X)$$ of $$G$$ on the vector space $$X=\F^n$$ is given by
\begin{equation*} g\to [v\to g\cdot v], \end{equation*}
where the "dot" in the expression $$g\cdot v$$ is ordinary multiplication of a matrix times a column vector. Show that this is indeed a group action.

5.

1. Show that $$\sim_G$$ is an equivalence relation.
2. Show that $$x\sim_G y$$ if and only if $$y\in \Orb(x)\text{.}$$
3. Explain why $$X/G$$ is a partition of $$X\text{.}$$

7.The projective linear group action on projective space.

Let $$V$$ be a vector space over a field $$\F$$ (in this course, the base field $$\F$$ is either the real numbers $$\R$$ or the complex numbers $$\C$$). The group $$\F^\ast$$ of nonzero elements in $$\F$$ acts on the set $$V\setminus \!\{0\}$$ of nonzero elements in $$V$$ by scalar multiplication, that is, by the map $$\alpha \to [v\to \alpha v]\text{.}$$ The set of orbits $$(V\setminus\!\{0\})/\F^\ast$$ is called the projectivization of $$V\text{,}$$ or simply projective space, and is denoted $$\Proj(V).$$
1. Let $$\sim_{\text{proj}}$$ denote the equivalence relation that defines the orbits $$(V\setminus \!\{0\})/\F^\ast\text{.}$$ Verify that $$\sim_{\text{proj}}$$ is given by $$x\sim_{\text{proj}} y$$ if and only if $$x=\alpha y$$ for some $$\alpha\in\F^\ast\text{.}$$
2. Verify that the group $$GL(V)$$ (the group of invertible linear transformations of $$V$$) acts on $$\Proj(V)$$ by
$$g\cdot [{v}] = [g({v})]\tag{2.5.1}$$
for $$g\in GL(V)$$ and $${v}\in V\setminus\!\{0\}\text{.}$$
3. Show that the kernel of the map $$GL(V)\to \Perm(\Proj(V))$$ given by (2.5.1) is the subgroup $$K=\{\alpha\Id\colon \alpha\in \F^\ast\}\text{.}$$
4. Conclude that the projective linear group $$PGL(V):=GL(V)/K$$ acts on $$\Proj(V)\text{.}$$
5. Show that $$\F^\ast$$ acts on $$GL(V)$$ by $$\alpha\cdot T=\alpha T\text{,}$$ and that $$PGL(V)\approx GL(V)/\F^\ast\text{.}$$
6. Let $$s\colon S^2 \to \extC$$ denote the stereographic projection (see (1.3.6)). Show that the map $$\Proj(\C^2)\to S^2$$ given by $$[(\alpha,\beta)]\to s^{-1}(\alpha/\beta)$$ if $$\beta\neq 0$$ and given by $$[(\alpha,\beta)]\to (0,0,1)$$ if $$\beta=0$$ is well-defined and is a bijection.