Group actions

In algebra, geometry and topology we often exploit the fact that important structures arise from families of morphisms that are indexed by a group. For example, rotations in the plane about the origin are indexed by the unimodular group of complex numbers; we say that this group acts on the plane and the orbit of a point at distance r from the origin is the circle of radius r.

We use in geometry the groups that act on subsets of $\mathbb{E} ^n$ while preserving Euclidean distances and angles; these are groups of isometries of $\mathbb{E} ^n.$ They form subgroups of matrix groups. The set of $n \times n$ nonsingular real matrices forms a group $GL (n,\mathbb{R} )$, often just written GL(n), the general linear group, under matrix multiplication. So does O(n), the subset consisting of orthogonal matrices, and its subset SO(n) consisting of those with determinant +1.

The Euclidean group E(n) consists of all isometries of Euclidean n-space $\mathbb{E} ^n$. Isometries can always be written as an ordered pair from $O(n) \times \mathbb{R} ^n$ with action on $\mathbb{E} ^n$ given by

$$(O(n) \times \mathbb{R} ^n ) \times \mathbb{E} ^n \longrighta... ... \mathbb{E} ^n : ((\alpha, u ), x ) \longmapsto \alpha (x) + u$$

and composition

$$(\alpha, u )(\beta, v ) = (\alpha \beta, \alpha (v) + u )\,.$$

Thus, topologically E(n) is the product $O(n) \times \mathbb{R} ^n$but algebraically it is not the product group. It is called a semidirect product of O(n) and $\mathbb{R} ^n$.

Definitions
A group G is said to act on a set (for example, a group, vector space, manifold, topological space) X on the left if there is a map (for example, homomorphism, linear, smooth, continuous)

$$\alpha : G \times X \longrightarrow X : (g,x) \longmapsto \alpha_g (x)$$

such that $\alpha_{g*h} (x) = \alpha_g (\alpha_h (x))$ and $\alpha_e (x) = x$ for all $x\in X$. Normally, we shall want each $\alpha_g : X \to X$ to be an isomorphism in the category for X; in this case, an action is the same as a representation of G in the automorphism group of X, or a representation on X. We sometimes abbreviate the notation to $g\cdot x$, especially when $\alpha$ is fixed for the duration of a discussion. There is a dual theory of actions on the right; we have to keep the concepts separate because every group acts on itself by its group operation, but it may be different on the right from on the left.

The orbit   of $x\in X$ under the action $\alpha$ of G is the set

$$G\cdot x = \{ \alpha_g (x) \mid g \in G\}\,.$$

It is easy to show that the orbits partition X, so they define an equivalence relation on X:

$$x\sim y \ \Longleftrightarrow \ \exists\, g\in G \mbox{ with } \alpha_g(x) = y\, .$$

The quotient object (set, space, etc.) is called the orbit space  and denoted by X/G.

The stabilizer  or isotropy subgroup  of x is defined to be the set

$$\mbox{stab}_G (x) = \{ g \in G \mid \alpha_g (x) = x \},$$

and it is always a subgroup of G.

The action is called transitive if for all $x,y \in X$ we can find $g \in G$ such that

$$\alpha_g (x) = y \qquad (\mbox{so also } \alpha_{g^{-1}} (y) = x)\, ,$$

free   if the only $\alpha_g$with a fixed point has g = e (the identity of G),
and effective   if

$$\alpha_g (x) = x \quad (\forall\, x \in X ) \ \Longrightarrow \ g = e.$$

Note that an action being transitive is equivalent to it having exactly one orbit, or to its orbit space being a singleton.

The situations of most practical interest are when:

• X is a subset of Euclidean space, a group or vector space--especially $\mathbb{S} ^n$, $\mathbb{E} ^n$ or $\mathbb{R} ^n;$

• G,X are topological groups , so each has a topology with respect to which its binary operation and the taking of inverses is continuous;

• G is a topological group and X is a topological space;

• G is a Lie group, so G has a differentiable structure with respect to which its binary operation and the taking of inverses is smooth, and X is a smooth manifold. Here smooth means all derivatives of all orders exist and are continuous. Important examples of Lie groups are $\mathbb{R} ^n,$ GL(n) and $\mathbb{S} ^1,$ where the differentiability arises from that of the underlying real functions.

Exercises on group actions

1.

$(\mathbb{Z} ,+)$ is a subgroup of $(\mathbb{R} , + )$.

2.

The symmetric group S_n of permutations of n objects is not abelian for n > 2.

3.

Find a group G consisting of four, $2\times 2$ real matrices such that Gacts on the plane $\mathbb{E} ^2.$For the case n = 2 find discrete subgroups G < E(2) such that $\mathbb{R} ^n /G$ is: (i) the cylinder; (ii) the torus.

4.

The general linear group $GL(n;\mathbb{R} )$ is not abelian if n > 1.

5.

Prove that GL(2) has a subgroup consisting of rotations in a plane

$$\left\{ \left( \begin{array}{cc}\cos\theta & -\sin\theta\\ ... ... \right) \left\vert \right. \ \theta\in \mathbb{R} \right\} .$$

This is actually SO(2), the special orthogonal group of $2\times 2$real matrices.

6.

Find an isomorphism

$$f:SO(2)\rightarrow \{z\in\mathbb{C} \vert \ \vert z\vert=1\}$$

and give its inverse.

7.

Prove that, for all elements a in group G, the map

$$c_a:G\rightarrow G:x\mapsto a^{-1}xa$$

is an automorphism; find the inverse of c_a.

8.

The group SO(2) of rotations in a plane acts on a sphere $\mathbb{S} ^2$as rotations of angles of longitude. The orbits are circles of latitude and the quotient space by this action is the interval [-1,1]. The action is neither transitive nor free, but it is effective.

9.

Prove that SO(2) defines a left action on $\mathbb{E} ^2$ by

$$\rho :SO(2)\times \mathbb{E} ^2 \rightarrow \mathbb{E} ^2:(A,p) \mapsto L_A p$$

where L_A p denotes matrix multiplication of the coordinate column vector p by the matrix A. To establish this you need to show that the map $\rho$is well-defined and that it satisfies two rules for all $p\in\mathbb{E} ^2$ and all $A,B\in SO(2),$ namely

Product

L_A (L_Bp)=L_AB p

Identity

L_I p=p

[In fact, the whole of the general linear group GL(2) acts on $\mathbb{E} ^2.$]

10.

Prove that the action $\rho$ is effective but neither free nor transitive. Find the orbits under this action of the points on the x-axis of $\mathbb{E} ^2.$

11.

Prove that the action $\rho$ preserves the scalar product; that is, for all $p, q\in \mathbb{E} ^2$ and all $A\in SO(2),$

$$L_Ap\cdot L_Aq = p\cdot q$$

Hence deduce that the action preserves Euclidean angles, lengths and areas.

12.

Show that

$$L_J=\left(\begin{array}{cc} 0 & -1\\ 1 & 0 \end{array} \right) \in SO(2)$$

and find the image under L_J of the unit square in the upper right quadrant of $\mathbb{E} ^2.$ [Hint: Check the edge vectors.] Find an element $K \in GL(2)$ with $K \not\in SO(2)$ and $\det K = -1.$This defines a linear map L_K; compare its effect on the unit square with the image found for L_J.

13.

It is clear that GL(3), which acts on $\mathbb{E} ^3,$ has a subgroup SO(3),consisting of $3\times 3$ real matrices having determinant +1. Find three distinct subgroups of SO(3), consisting of rotations around the three coordinate axes, respectively, by finding three group homomorphisms $SO(2)\rightarrow SO(3)$ with trivial kernels.

14.

Use the subgroups of SO(3) found in the previous exercise, and the parametric equation for the equator of $\mathbb{S} ^2,$ to show how any other great circle on $\mathbb{S} ^2$ can be found by appropriate combinations of rotations of the equator.

15.

Find two matrices, R₁ and R₂ from SO(3)which represent, respectively, rotation by $\pi/3$ about the y-axis and rotation by $\pi/4$ about the z-axis; each rotation must be in a right-hand-screw sense in the positive direction of its axis. Find the product matrix R₁R₂ and show that its transpose is its inverse.