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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. \(\Orb(x)=\{gxg^{-1}\colon g\in G\}\text{,}\) \(\Stab(x)=C(x)\) (the centralizer of \(x\))

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.

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.

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 \(X\text{.}\)

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{2.5.1} \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 (2.5.1) 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.
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