## Section2.5Group actions

### Subsection2.5.1

###### 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  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  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.

Other notations for $(\phi(g))(x)$ are $g(x)\text{,}$ $g\cdot x\text{,}$ and $g.x\text{.}$
It must be proved that $\Stab(x)$ is indeed a subgroup of $G\text{.}$ See Checkpoint 2.5.3 below.

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 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. $\Z$ acts on $\R$ by translation, that is, by $(\phi(n))(x)= x+n\text{.}$ What is the orbit of $1\text{?}$ What is the orbit of $\pi\text{?}$ What is the stabilizer of $1\text{?}$ What is the stabilizer of $\pi\text{?}$
3. Any group $G$ acts on itself by conjugation, that is, by $(\phi(g))(x)=gxg^{-1}=C_g(x)$ (see Exercise 2.4.2.9). 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. $\Orb(1)=\Z\text{,}$ $\Orb(\pi)=\{\pi+n\colon n\in \Z\}\text{,}$ $\Stab(1)=\{0\}=\Stab(\pi)$
3. $\Orb(x)=\{gxg^-1\colon g\in G\}\text{,}$ $\Stab(x)=C(x)$ (the centralizer of $x$)

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

### Exercises2.5.3Exercises

###### 1.Actions of a group on itself.

Let $G$ be a group. Here are two 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.

###### 2.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 2.5.3.1 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.

###### 3.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.

###### 6.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})]\label{glnprojaction}\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. 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.