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# 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 be a set and let be a group. A (left) action of on is a function such that
• If is the indentity element, then for all .
• If and are in , then for all in .

Exercise 2.1 (00)   Show that the map    Bij is a homomorphism of groups. Use this to redefine the action of a group on a set.

Definition 2.2   An action of a group on a set is said to be transitive if whenever and are in , there exists an element such that .

The following construction furnishes all examples of transitive group actions: if is a group and is a subgroup, we define the equivalence relation (or ) if . The quotient is the set of right cosets of which is also denoted by .

Exercise 2.2   Show that the action defined by is transitive.

Definition 2.3   Let be a transitive action of a group on a set . If , we define the isotropy subgroup of , , as the set of all which fix (i.e., ).

Exercise 2.3   Let be a transitive action of a group on a set and let be an element of . Find a canonical bijection between and .

Exercise 2.4 (05)   If is the group of orthogonal matrices and denotes the unit sphere in , then there is a natural action

defined by .
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'' .
3. Write the sphere as a quotient space of .
4. * What is the dimension of ?
5. * Show that orthogonal transformations are the only elements of Bij 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 defined by . Show that this action is transitive and write the sphere as a quotient space of .

Exercise 2.6 (20)   Find transitive actions of the unitary group or the special unitary group on the -dimensional unit sphere and write as a quotient space of and . * Use this to compute the dimension of and .

Exercise 2.7 (15)   The -dimensional Euclidean group is the set provided with the product .
1. Verify that is a group.
2. Show that the map defined by (rotate, then translate) is a transitive action.
3. * Show that Euclidean transformations are the only elements of Bij that preserve distances.

Exercise 2.8 (*20)   The -dimensional affine group is the set provided with the product . Show that acts transitively on the set of ellipses (or parabolas, or hyperbolas) on the plane.

Next: Bibliography Up: Groups and Their Actions Previous: Important examples from linear
Juan Carlos Alvarez 2000-10-27