Groups were first studied as objects acting on sets. For example, we can consider the group of rotations of a regular polygon. Eventually, we wish to consider a specific type of group acting on the collection of rational functions
over a field
. Today, we review various definitions and properties by focusing exclusively on dihedral, symmetric, alternating, and matrix groups.
Definitions of Groups
Let be a set, and assume that there is some composition “
” on
; that is, given
there is a unique element
. We say that
is a group under
if the following four properties hold:
- Closure: Given
the composition
.
- Associativity: Given
we have
. (This is to make sure the expression
is unambiguous.)
- Identity: There is a unique element
such that for all
we have
. (We call
the identity element under
.)
- Inverses: Given
there is a unique element
such that
. (We call
the inverse of
, and sometimes write
or
depending on the operation
.)
We say that is an abelian group if (Closure) — (Inverses) hold, and additionally
- Commutativity: Given
we have
.
We say is finitely generated if there exists a finite set
such that, for any
, there exist integers
such that
. The set
is called a generating set for the group
; this set is not unique. We often write
, where
is the collection of ways to express
. This is a presentation for the group
; it is also not unique.
Dihedral Groups
We give a fundamental example of a group which is not abelian.
Theorem.
For each positive integer, let the dihedral group
denote the set of symmetries of the regular
-gon
.
- The group
has
elements.
- It can be generated by just two elements, namely the rotation
of
radians counterclockwise and a reflection
about the line
of slope
.
- In particular, a presentation for the dihedral group is
.
Proof: First, let’s write down some obvious symmetries of the -gon. We have a rotation
as well as a reflection
about the line
. We can express both in terms of matrices:
and
. It is easy to check that
.
Now we can count the number of symmetries. Any symmetry must be a rigid rotation of the regular
-gon, so it is uniquely determined by where it sends the two vertices
and
. Say that
. Since
is adjacent to
, it must go to
. The symmetry
, while
and
. Hence either
or
. There are
such choices.
Morphisms between Groups
Say that and
are groups under the composition laws
and
, respectively. We say that a well-defined map
is a group homomorphism if it preserves the group operations, that is,
. The set of group homomorphisms from
to
is denoted by
. We say that a group homomorphism is an isomorphism if it is a bijection. If
is indeed an isomorphism, we say that
and
are isomorphic and write
. Similarly, a group homomorphism
is called an endomorphism; a bijective endomorphism is called an automorphism. We denote these sets by
and
, respectively.
Proposition.
Let,
, and
be groups under the composition laws
,
, and
, respectively. Fix
and
.
Proof: Say that and
. Clearly
is a well-defined map. For
we have
Hence is also a group homomorphism.
To show is a group under composition, we must show that four axioms are satisfied. To see why (Closure) holds, note that the composition
for any
. The composition of two bijections is also a bijection because
. Property (Associativity) holds for the composition of maps.
To see why (Identity) holds, consider the trivial map . We have
so that
is a group homomorphism, and it is clear that
is a bijection. To see why (Inverses) holds, say that
and
. Then we have
showing that is a also group homomorphism.
We show that is a subgroup of
. To see why (Closure) holds, observe that
so that
for any
. Property (Associativity) holds because it holds for
. To see why (Identity) holds, observe that
for any
, so that
. To see why (Inverses) holds, note that
whenever
so that the inverse of
is
. In particular, if
then
as well.
Finally we show that is a subgroup of
. To see why (Closure) holds, observe that
whenever
, so that
as well. Property (Associativity) holds because it holds for
. To see why (Identity) holds, recall that
. To see why (Inverses) holds, recall that
.
Group Actions
Let be a group under a composition law
with identity
. A group action by
on a set
is a well-defined map
, which we write as
, satisfying the following axioms:
- Associativity:
for all
and
.
- Identity:
for all
.
Denote the permutation group as the set of all permutations
, that is, well-defined maps which are both injective and surjective. When
acts on a set
, we may think of
as being “contained” in
.
Proposition.
Letbe a group acting on a set
.
- For each
, the map
defined by
is a permutation.
- The map
defined by
is a group homomorphism.
- If
has
elements, then
has
elements.
Proof: First observe that for all
and
so that
. Hence
, so that
is indeed a well-defined bijection. The map
is defined by
. Since
, we see that
is indeed a group homomorphism.
Let . Since
, there
possibilities for
. By assumption,
is injective, so
. Hence there are
possibilities for
. Continuing in this fashion, we see that there are at most
possibilities for
. We have actually counted the number of injective maps
. Each such map gives rearrangements for the elements of
, so each is really an surjective map. Hence of these
possibilities is really a bijection.
The map defined by
is called the permutation representation of the associated action on
. If the kernel of this group homomorphism is trivial, that is,
, then we say that
acts faithfully on
.
Symmetric and Alternating Groups
The symmetric group of the set is simply denoted by
, and is called the symmetric group of degree
. This group acts on the collection
of
distinct variables by
. We will use this action to define a normal subgroup of
.
Proposition.
The mapdefined by
is a group homomorphism.
Proof:We have the identity
for all .
The image is called the sign of a permutation
. We say that
is even (odd, respectively) if
(
, respectively). The kernel of this group homomorphism is defined as the alternating group of degree
:
. Note that
is a subgroup of
.
Matrix Groups
We generalize the idea behind dihedral groups by considering matrices in a more general setting. Let denote either
,
, or
. Recall that a matrix over
is an
array, that is, a grid with
rows and
columns in the form
where
. We will use shorthand notation to write
to denote that
is the entry in the
th row and
th column. The set of all such matrices is denoted by
.
We will be interested in the set square matrices over , that is,
. The sum and product of two square matrices is defined as
Proposition.
Letdenote either
,
, or
. Then set
consisting of those invertible
a group under matrix multiplication.
Proof:We have to verify four axioms. To show (Closure), note that for any
so that the product of invertible square matrices is again an invertible square matrix. To show (Associativity), denote
,
, and
as three elements from
. We have
To show (Identity), note that the identity matrix
is an invertible square matrix. To show (Inverses), recall that
, so that the inverse of an invertible square matrix is again an invertible square matrix.
The set is called the general linear group of degree
over
. When
, we have
as a group under multiplication. Hence,
is a natural generalization. There is another way to define this group. Recall that a matrix
is invertible if and only if its determinant is nonzero:
. Since the determinant is a multiplicative map, that is,
, we may view the determinant as a group homomorphism
. The special linear group of degree
is the kernel of this homomorphism:
. Note that
is a subgroup of
.
Fields and Extensions
A triple is said to be a field if the following three axioms hold:
- Additive Group: The pair
is an additive group with identity 0.
- Multiplicative Group: The pair
is a multiplicative group, where
.
- Distributivity: For all
we have the distributive law
.
The sets ,
, are
are each examples of fields. A subset
is called a subfield if
is also a field; we also say that
is an extension of
.
Given a variable , a polynomial over
is a sum
where each
. If
, we say that
is the degree of the polynomial
. A rational function over
is an expression
which is the ratio of two polynomials
and
over
. If
and
are relatively prime, that is, the
determinant
of the Sylvester Matrix is nonzero, we say that
is the larger of the degrees of the polynomials in the numerator and the denominator. The collection of all such rational functions is the set
. It is straight-forward to verify that
is indeed a field and hence an extension of
. We will consider those fields
which are transcendental over
, that is,
is an extension of
. Note that if
is a rational function from
, then the collection
of rational functions in
is an extension of the collection
of rational functions in
. Whenever
has
, we say that
is an algebraic extension of
of degree
.
Möbius Transformations
Let denote either
,
, or
. The collection of rational functions
which have
are called Möbius Transformations, named in honor of the German mathematician August Ferdinand Möbius (1790 — 1868). That is, they are in the form
where
. We denote this collection by
.
Proposition.
Letdenote either
,
, or
.
forms a group under composition.
acts on
via
.
We explain why the notation . Indeed, each
acts on
. Explicitly,
maps to the Möbius transformation
. Such a map is both injective and surjective, and hence a permutation of the projective line
.
Proof:We explain why the forms a group. Define
as composition, that is,
where
for
. It is easy to check that this is a well-defined operation. In fact, this shows that the well-defined map
which sends
to
is a surjection and preserves the composition laws:
. But the image
, so it is clear that
must be a group.
Consider the map which sends a pair
to the rational function
. Property (Associativity) holds because
for all and
. Property (Identity) holds because the function
acts trivially:
. Hence
is indeed a group action.
.
Since Möbius transformations act on rational functions
, we define the Galois group of
as the subgroup
. We are motivated by a fundamental question.
Motivating Questions.
What are the possible groups which the Galois groupcan be? That is, given a finite group
, can we find a rational function
such that
?
We will focus on these questions more in the next lecture.
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