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Introduction to Groups and Geometries

David W. Lyons

Section 2.5 Group actions

Definition 2.5.1. 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.

Checkpoint 2.5.2.

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 \(x\text{.}\)
Answer.
  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. \(\Orb(x)=\{gxg^{-1}\colon g\in G\}\text{,}\) the stabilizer of \(x\) is the centralizer subgroup \(C(x)\) (see Exercise 2.3.5)

Checkpoint 2.5.3.

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

Definition 2.5.4. 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.

Checkpoint 2.5.5.

Exercises Exercises

1. The sign of a permutation.

Let \(n\) be a positive integer, and let \(\Delta_n\) be the polynomial
\begin{equation*} \Delta_n = \prod_{1\leq i\lt j\leq n} (x_i-x_j) \end{equation*}
in the variables \(x_1,x_2,\ldots,x_n\text{.}\) For example, we have \(\Delta_3 = (x_1-x_2)(x_1-x_3)(x_2-x_3)\text{.}\) Given a permutation \(\alpha\) in \(S_n\text{,}\) let \(\alpha \Delta_n\) be the polynomial
\begin{equation*} \alpha \Delta_n = \prod_{1\leq i\lt j\leq n} (x_{\alpha(i)}-x_{\alpha(j)}). \end{equation*}
In the exercises below, you will show that \(\alpha\Delta_n=\pm \Delta_n\) for all \(\alpha\) in \(S_n\text{.}\) This allows us to define the sign of a permutation \(\alpha\text{,}\) denoted \(\sgn(\alpha)\), to be \(+1\) or \(-1\) according to whether \(\alpha\Delta_n=\Delta_n\) or \(\alpha\Delta_n=-\Delta_n\text{,}\) respectively.
\begin{equation*} \sgn(\alpha) = \begin{cases} +1 \amp \text{if } \alpha\Delta_n = \Delta_n\\ -1 \amp \text{if } \alpha\Delta_n = -\Delta_n \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 expressions for \(\Delta_4\text{,}\) \((134)\Delta_4\text{,}\) and \((1324)\Delta_4\) (write all the factors). All three expressions consist of the same factors, possibly times \(\pm 1\text{.}\) Identify which factors of \((134)\Delta_4\) and \((1324)\Delta_4\) have the opposite sign of the corresponding factor in \(\Delta_4.\) Use your expressions to find \(\sgn(134)\) and \(\sgn(1324)\text{.}\)
  2. Generalize the examples in part (a) above to show that \(\alpha \Delta_n = \pm \Delta_n\) for all \(\alpha\) in \(S_n\text{.}\) This justifies the definition of the sign of a permutation.
  3. Show that \(\tau\Delta_n=-\Delta_n\) for any transposition \(\tau\) in \(S_n\text{.}\) Suggestion: Let \(\tau=(ab)\) be a transposition with \(a\lt b\text{.}\) Count the number of factors \((x_i-x_j)\) of \(\Delta_n\) such that \(x_{\tau(i)}-x_{\tau(j)} = -(x_i-x_j)\text{.}\)
  4. Define \(\alpha(-\Delta_n)\) by \(\alpha(-\Delta_n)=-\alpha\Delta_n\) for \(\alpha\) in \(S_n\text{.}\) Show that
    \begin{equation} (\alpha \beta)\Delta_n = \alpha(\beta\Delta_n)\tag{2.5.1} \end{equation}
    for all \(\alpha,\beta\) in \(S_n\text{.}\)
  5. Let \(\alpha\) be an element of \(S_n\text{.}\) By Exercise 2.2.9, \(\alpha\) may be written as a product of transpositions \(\alpha=\tau_1\tau_2\cdots\tau_k\text{.}\) Use (2.5.1) to show that \(\sgn(\alpha)=(-1)^k\text{.}\) Consequence: if \(\alpha\) is expressible as a product of an even number of transpositions, then \(\alpha\) is an even permutation. Further, 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”.)
  6. Show that \(\phi(\alpha)(\pm \Delta_n) = \pm \alpha\Delta_n\) defines a group action \(\phi\colon S_n \to \Perm(X)\) of the group \(S_n\) on the set \(X=\{\Delta_n,-\Delta_n\}\text{.}\)
  7. Show that the sign function \(\sgn\colon S_n\to C_2\) is a homomorphism of groups.
  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.
Hint.
For part (d): Both sides of (2.5.1) are equal to \(\prod_{i\lt j} \left(x_{\alpha\beta(i)}-x_{\alpha\beta(j)}\right)\text{.}\)

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 \(L\) 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 \(R\) is given by
\begin{equation*} g\to R_g \end{equation*}
where \(R_g\) is given by \(R_g(h)=hg^{-1}\text{.}\) Conjugation \(C\) 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.

Prove Proposition 2.5.6.
  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 \(X\text{.}\)

6.

Prove The Orbit-Stabilizer Theorem (Theorem 2.5.7).
Hint.
Let \(x\in X\text{,}\) and define a map \(f_x\colon G\to X\) given by \(f_x(g)= gx\text{.}\) Apply Fact 1.4.5 to get a one-to-one correspondence
\begin{equation*} G/\!\!\sim_{f_x} \leftrightarrow f_x(G)=\Orb(x). \end{equation*}
Then use Proposition 2.3.9 to show that \(G/\!\!\sim_{f_x} = G/\Stab(x)\text{.}\) Whatever proof method you use, you must address the issue of well-definedness of one of the maps \(g\Stab(x)\to gx\) or \(gx \to g\Stab(x)\text{.}\)

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
    \begin{equation} g\cdot [{v}] = [g({v})]\tag{2.5.2} \end{equation}
    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 (2.5.2) 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 (1.3.4)). 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.
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