Actions of groups on sets

In geometry groups are often defined as subgroups of the group of bijections of a set. However, an abstract definition of what it means for a group to act on a set is still useful.

Definition 2.1   Let $X$ be a set and let $(G,*)$ be a group. A (left) action of $G$ on $X$ is a function $\mu : G \times X \rightarrow X$ such that

• If $e \in G$ is the indentity element, then $\mu(e,x) = x$ for all $x \in X$.
• If $g$ and $h$ are in $G$, then $\mu(g,\mu(h,x)) = \mu(g*h,x)$ for all $x$ in $X$.

Exercise 2.1 (00)   Show that the map $g \mapsto \mu(g, \cdot) \in$   Bij$(X)$ is a homomorphism of groups. Use this to redefine the action of a group on a set.

Definition 2.2   An action $\mu$ of a group $(G,*)$ on a set $X$ is said to be transitive if whenever $x$ and $y$ are in $X$, there exists an element $g \in G$ such that $\mu(g,x) = y$.

The following construction furnishes all examples of transitive group actions: if $G$ is a group and $H \subset G$ is a subgroup, we define the equivalence relation $g \sim g'$ (or $[g] = [g']$) if ${g'}^{-1}*g \in H$. The quotient $G/\sim$ is the set of right cosets of $H$ which is also denoted by $G/H$.

Exercise 2.2   Show that the action $\mu : G \times G/H \rightarrow G/H$ defined by $\mu(g,[f]) = [g*f]$ is transitive.

Definition 2.3   Let $\mu : G \times X \rightarrow X$ be a transitive action of a group $G$ on a set $X$. If $x \in X$, we define the isotropy subgroup of $x$, $G_{x}$, as the set of all $g \in G$ which fix $x$ (i.e., $\mu(g,x) = x$).

Exercise 2.3   Let $\mu : G \times X \rightarrow X$ be a transitive action of a group $G$ on a set $X$ and let $x$ be an element of $X$. Find a canonical bijection between $X$ and $G/G_{x}$.

Exercise 2.4 (05)   If $O(n)$ is the group of orthogonal $n \times n$ matrices and $S^{n-1}$ denotes the unit sphere in ${\Bbb R}^n$, then there is a natural action

$$\mu : O(n) \times S^{n-1} \longrightarrow S^{n-1}$$

defined by $\mu(A,x) := Ax$.

1. Verify that this is an action and show that it is transitive.
2. Indentify the subgroup of all orthogonal transformations fixing the ``north pole'' $(0,\dots,0,1)$.
3. Write the sphere as a quotient space of $O(n)$.
4. * What is the dimension of $O(n)$?
5. * Show that orthogonal transformations are the only elements of Bij$(S^{n-1})$ that preserve distances.

Sometimes a set admits many different transtive actions and can be represented as a group quotient is different ways.

Exercise 2.5 (05)   Consider the action $\mu : GL(n,{\Bbb R}) \times S^{n-1} \rightarrow S^{n-1}$ defined by $\mu(A,x) = Ax/\Vert Ax\Vert$. Show that this action is transitive and write the sphere as a quotient space of $GL(n,{\Bbb R})$.

Exercise 2.6 (20)   Find transitive actions of the unitary group $U(n)$ or the special unitary group $SU(n)$ on the $(2n-1)$-dimensional unit sphere and write $S^{2n-1}$ as a quotient space of $U(n)$ and $SU(n)$. * Use this to compute the dimension of $U(n)$ and $SU(n)$.

Exercise 2.7 (15)   The $n$-dimensional Euclidean group $E(n)$ is the set $O(n) \times {\Bbb R}^n$ provided with the product $(A,x) * (B,y) = (AB, x+ Ay)$.

1. Verify that $E(n)$ is a group.
2. Show that the map $\mu: E(n) \times {\Bbb R}^n \rightarrow {\Bbb R}^n$ defined by $\mu((A,x),y) = Ay + x$ (rotate, then translate) is a transitive action.
3. * Show that Euclidean transformations are the only elements of Bij$({\Bbb R}^n)$ that preserve distances.

Exercise 2.8 (*20)   The $n$-dimensional affine group $A(n)$ is the set $GL(n,{\Bbb R}) \times {\Bbb R}^n$ provided with the product $(A,x) * (B,y) = (AB, x+ Ay)$. Show that $A(2)$ acts transitively on the set of ellipses (or parabolas, or hyperbolas) on the plane.