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.
We give a fundamental example of a group which is not abelian.
- 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.
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 .
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 .
Let be 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.
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 .
The map defined 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 .
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 .
Let denote 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 .
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 .
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 .
Let denote 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.
What are the possible groups which the Galois group can 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.