2.6 Group actions

A left action of a group $G$ on a set X is a homomorphism $\phi$ of $G$ into the group of transformations of $X$

$$\phi :G\to \mathrm{Transf}(X).$$

It is common to write $\phi (g)(x)$ simply as gx, a notation that makes it possible to write ghx in place of $(gh)x$ , since

$$(gh)x=\phi (gh)(x)=\phi (g)\phi (h)(x)=\phi (g)(hx)=g(hx).$$

A left action φ ofG on ${\mathbb{R}}^n$ all ofwhose images are linear transformations is a homomorphism

$$\varphi :G\to GL(n,\mathbb{R}),$$

and is called an n-dimensional representation of G. Similarly, a homomorphism $\varphi :G\to$ $GL(n,\mathbb{C})$ is called a complex n-dimensional representation of G.

An anti-homomorphism is defined as a map $\rho :G$ Transf(X) with the property

$$\rho (gh)=\rho (h)\rho (g).$$

It can give rise to a right action $xg=\rho (g)(x)$ , a notation that is consistent with writing $xgh$ in place of $x(gh)=(xg)h$

Exercise

$\mathrm{If}\phi :G\to$ H is a homomorphism show that the map $\rho :GH$ defined by $\rho (g)=\phi (g^{-1})$ is an anti-homomorphism.

Let G be a group having a left action on X. The orbit Gx of a point $x\in X$ is the set of all points that can be reached from x by this action,

$$Gx=\{gx\mid g\in G\}.$$

We say the action of $G$ on X is transitive if the whole of X is the orbit of some point in $X,$

$$\exists x\in X\text{ such that }X=Gx.$$

In this case any pair ofelements $y,z\in X$ can be connected by the action ofa group element, for if $y=gx$ and $z=hx$ then $z=g^{\prime }y$ where $g^{\prime }=h{g}^{-1}$ . Hence $X=Gy$ for all $y\in X$

If x is any point of $X,$ define the isotropy group of x to be

$$G_x=\{g\mid gx=x\}.$$

If $gx=x⟹g={\mathrm{id}}_X$ the action of G on X is said to be free. In this case the isotropy group is trivial, $G_x=\{{\mathrm{id}}_X\}$ , for every point $x\in X$

Exercise

Show that $G_x$ forms a subgroup of $G.$

If $x\in X$ and h, $h^{\prime }\in G$ then

$$hx=h^{\prime }x⟹h^{-1}{h}^{\prime }\in G_x⟹h^{\prime }\in h{G}_x.$$

If G is a finite group, we denote the number ofpoints in any subset S by S . Since $h{G}_x$ is a left coset of the subgroup $G_x$ and from the proof of Lagrange’s theorem 2.3 all cosets have the same number of elements, there must be precisely $|G_x|$ group elements that map x to any point y of its orbit Gx. Hence

$$|G|=|Gx||G_x|.\text{\hspace{1em}(2.16)}$$

Example 2.23

The cyclic group of order 2, ${\mathbb{Z}}_2=\{e,a\}$ where $a^2=e$, acts on the real numbers $\mathbb{R}$ by

$$ex=x,ax=-x.$$

The orbit of any point $x\ne 0$ is ${\mathbb{Z}}_{2}x=\{x,-x\}$ , while ${\mathbb{Z}}_{2}0=\{0\}$ . This action is not trans itive. The isotropy group of the origin is the whole o ${\mathbb{Z}}_2$ , while for any other point it is e . It is a simple matter to check Eq. (2.16) separately for $x=0$ and $x\ne 0$

Example 2.24

The additive group of reals R acts on the complex plane C by

$$\theta :z\mapsto z{e}^{i\theta }.$$

The orbit of any $z\ne 0$ is the circle centred 0, radius $r=|z|$ . The action is not transitive since circles of different radius are disjoint. The isotropy group of any $z\ne 0$ is the set of real numbers of the form $\theta =2\pi n$ where $n\in \mathbb{Z}$ . Hence the isotropy group ${\mathbb{R}}_z$ for $z\ne 0$ is isomorphic to $\mathbb{Z},$ the additive group of integers. On the other hand the isotropy group of $z=0$ is all of R.

Example 2.25

A group $G$ acts on itself by left translation

$$g:h\mapsto L_{g}h=gh.$$

This action is clearly transitive since any element $g^{\prime }$ can be reached from any other $g$ by a left translation,

$$g^{\prime }=L_{g^{\prime }{g}^{-1}}g.$$

Any subgroup $H\subseteq G$ also acts on $G$ by left translation. The orbit of any group element g under this action is the right coset $Hg$ containing g. Similarly, under the right action of H on $G$ defined by right translation $R_h:g\mapsto gh$ , the orbits are the left cosets $gH$ . These actions are not transitive in general.

Example 2.26

The process ofconjugation by an element ${}^{g,}$ defined in Eq. (2.14), is a left action of the group $G$ on itself since the map $g\mapsto C_g$ is a homomorphism,

$$C_{gh}a=(gh)a(gh{)}^{-1}=gha{h}^{-1}{g}^{-1}=C_g{C}_{h}a,$$

where we have written $C_{g}a$ for $C_g(a)$ . The orbits under the action ofconjugation are precisely the conjugacy classes. By Eq. (2.16) it follows that if G is a finite group then the numbe of elements in any conjugacy class, being an orbit under an action of $G_{:}$ , is a divisor of the order of the group G .

If G has a left action on a set X and if x and y are any pair of points in X in the same orbit, such that $y=hx$ for some $h\in G$ , then their isotropy groups are conjugate to each other,

$$G_y=G_{hx}=h{G}_x{h}^{-1}.\text{\hspace{1em}(2.17)}$$

For, let $g\in G_y.$ , so that $gy=y$ . Since $y=hx$ it follows on applying $h^{-1}$ that $h^{-1}ghx=x$ Hence $h^{-1}gh\in G_x$ , or equivalently $g\in h{G}_x{h}^{-1}$ . The converse, that $h{G}_x{h}^{-1}\subseteq G_y$ , is straightforward: for any $g\in h{G}_x{h}^{-1}$ , we have that

$$gy=ghx=h{g}^{\prime }{h}^{-1}hx\text{ where }{g}^{\prime }\in G_x,$$

whence $gy=hx=y$ and $g\in G_y$ . Thus the isotropy groups of x and $y$ are isomorphic since they are conjugate to each other, and are related by an inner automorphism. If the action of G on X is transitive it follows that the isotropy groups of any pair of points x and y are isomorphic to each other.

Exercise

Under what circumstances is the action of conjugation by an element g on a group G transitive?

Problem

Problem 2.19

If H is any subgroup of a group $G$ define the action of G on the set of left cosets $G/H$ by $g:g^{\prime }H\mapsto g{g}^{\prime }H$

(a) Show that this is always a transitive action of G on $G/H$.

(b) Let G have a transitive left action on a set X, and set $H=G_x$ to be the isotropy group of any point x. Show that the map $i:G/H\to X$ defined by $i(gH)=gx$ is well-defined, one-to-one and onto.

(c) Show that the left action of $G$ on X can be identified with the action of G on $G/H$ defined in (a).

(d) Show that the group ofproper orthogonal transformations $SO(3)$ acts transitively on the 2-sphere $S^2$

$$S^2=\{(x,y,z)|r^2=x^2+y^2+z^2=1\}=\{\mathbf{r}|r^2={\mathbf{r}}^T\mathbf{r}=1\},$$

where r is a column vector having real components x, y, z. Show that the isotropy group of any point r is isomorphic to $SO(2)$ , and find a bijective correspondence between the coset space $SO(3)/SO(2)$ and the 2-sphere $S^2$ such that $SO(3)$ has identical left action on these two spaces.