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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.

**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

.

- Verify that this is an action and show that it is transitive.
- Indentify the subgroup of all orthogonal transformations fixing the
``north pole''
.
- Write the sphere as a quotient space of .
- * What is the dimension of ?
- * 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 2.8** (*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