Groups

Group theory, which studies a type of algebraic structures called group, is the very central and fundamental concept of algebra. By studying group theory, we build up some concepts and proofs with which we could use to drill deeper into the mathematical world.

Groups

A group is a set equipped with a binary operation ¹ such that:

1. (Closed) .
2. (Identity) .
3. (Associative) .
4. (Invertible) .

For convenience, we usually omit the binary operation of group and abbreviate the group as its set notation , if which binary operation is applied on the group is clear under their current context.

For each group , if operations are commutative that , then the group is called an abelian group.

A group is said to be finite when the set is finite. When we meet propositions related to the size of a finite group, we usually refer to a group of order to provide more details or constraints, or refer to a group of infinite order for the case of infinite group.

As you might have noticed, we denote the invert element of as using power. In fact, the definition is compatible with natural power such that , and where is positive integer. It won't be hard to find out that . We will use this notation from now on ².

Group is really common, noted that is a group, the reduced residue system of equipped with the multiply operation is a group, and the most favourable computer graphic homogeneous matrix and its multiply operation is also a group ³.

On the other hand, group is so common that most of us hardly notice there're some non-trivial properties in them. And only by revealing those non-trivial properties will we be getting to understand what algebra is.

Uniqueness

For each group, the identity element is unique. To prove, let's assume that there're two distinct identities and in the group, and consider their product . Since is identity, there will be . Since is identity, there will be . So there will be , which is a contradiction to their distinction.

For each element in the group, its inverse element is unique. Similarly, let's assume there're two distinct invert elements and , and consider the result of . Let's associate the first two elements, there will be . And Let's associate the last two elements, there will be . So there will be , which is a contradiction to their distinction.

Bijectivity

As we have noticed in the reduced residue system that multiplying by an element repremutates all the elements inside, or in other word, creates a bijective automorphism on the reduced residue system. So naturally we will hope it is also true for a group.

Let's consider the left multiplication ⁴ defined by . To show its injectivity, assume , left multiplying by yields , which implies and is a contradiction to . To show its surjectivity, let's randomly take from , and obviously there's (at least) a preimage which must be in , otherwise it will be a contraction to 's closed property. Now we've shown that is a bijective automorphism on .

The right multiplication defined by can also be shown to be a bijective automorphism in an analogous way.

"Square root"

It's good to know that iff . Its sufficiency is obvious, while its neccesarity requires proof. To prove, let's assume that , multiplying by yields , which is a contradiction.

This is a recurrence of uniqueness of group structure defined by a group set and its binary operation . Due to iff , when there're multiple group structures on set they must at least share the same identity element.

Subgroups

A group might also have some substructures: it won't be hard to find out all even number equipped with add operation also forms a group, which is a substructure of . Same fact also holds for where .

Given a group , if and is also a group, then group is called a subgroup of . We also denote such relation between subgroup and group as .

Things will be getting more interesting when it comes to finite groups, which enables us to consider the numerical relationships between groups and subgroups. Like given a group and its subgroup , is it possible that and ? You'll soon notice that the answer to such kind of problem is non-trivial.

Element constraints

Let's consider what elements in should also be in .

Starting from the identity element in , just in case of that it is different from in , we denote it as . Since is also a group, , while and yields , which is a contradiction. So it is only possible that .

Given an element in , it should be invertible so and is also in .

Given elements in , their binary operation need to be closed so and should also be in .

And since we are selecting elements from to under the associative binary operation , the property of associative holds automatically as long as enough elements are selected so that the operation is closed. So it won't be hard to find out and prove that a group is subgroup of iff:

1. If is the identity element of , .
2. .
3. .

It won't be hard to verify that is the minimum group (since it's only possible that , we always omit the binary operation while referring to it), and is the subgroup of all groups. So as a subgroup is also referred as trivial subgroup.

Lagrange's theorem

Lagrange's theorem states that when , there's a way to partition into disjoint subsets such that there're bijections between 's and each subset's elements. And if is finite, must divides .

In order to depict what's happening precisely, we define the left coset of as . By the bijectivity of 's left multiplication, is a bijection that can map 's elements into 's. We'll show it's either the case that or the case that .

The previous case is trivially true since is a group and its binary operation must be closed.

The latter case could be proved by contradiction. Let's assume that , and . Multiplying by yields . Obviously , and since , there must be , which is a contradiction to ⁵.

Let be finite, imagine we can iterate the process on such that we first remove from , where the remaining, element in . And if is still not empty, we randomly choose an element from it and remove . And since it's disjoint with , . Noted that we can denote so that , we intuitively expect there exists a process such that so that and divides , but this requires cosets are also pairwisely disjoint, not only disjoint with . And we are just about to show that.

Inspired by the proof of and are identical or disjoint, as you can see, given element , if , then is obviously in , given that it's the image of under . Conversely since is group, , so is also in under . And if , then is also in . We've effectively found an equivalence relation defined by on such that . It won't be hard to find that . And due to 's being equivalence relationship, , which concludes the proof of that distinct cosets of are pairwisely disjoint.

And now you know, it's impossible for and to be true, Lagrange's theorem does reveal some non-trivial properties of group when it has a subgroup.

It's possible for subgroup to partition into finite number of cosets despite and 's being infite, like partitions into cosets. And sometimes we may meet propositions involving merely the number of cosets, and simply writing does not make sense when both and are infinite. In case of such circumstance, we define the index of in as the number of cosets of in , and denote it as . And obviusly when is finite, and coincide.

Quotient groups and normal subgroups

Consider the addition operation on and its congruence modulo-: given that , there's , where might be another modulo- equivalence of , or their remainders. And after learning the concept of subgroup and Lagrange's theorem, it won't be hard for us to find out that is a subgroup of , partitioning into different cosets such that means , so do and .

So the addition on 's congruence modulo- can also be viewed as the addition operation on coset: taking arbitrary from coset , and from coset , their result always lies in , which is another coset. Since the property is applicable to every element in some coset , we elevate the property of elements to the property of their corresponding cosets, and can restate the condition as , so the coset can be viewed as sum of the two cosets in this way.

Inspired by the case study above, we generalize such inter-coset property for a group with subgroup . Under the group's binary operator , iff . And it won't be hard to show the cosets forms a group under the group's natural operator if the property above holds for all :

1. (Closed): , obviously , and are valid coset, inherited from the 's closed property.
2. (Identity): is the identity of the group of cosets, since , there's and .
3. (Associative): Since , with proper association appied to there'll be and , and by the associative property of there will be .
4. (Invertible): Since , .

For convenience, if is a subgroup of such that holds, we will also denote the set of cosets as instead of , where is the equivalence that iff . And the group formed in such way is called a quotient group.

Obviously such generalization sheds light on constructing some congruence-like group structure from a group and its subgroup with specified property, and exhibits some beauty of harmony.

However, we are not sure about whether is applicable to arbitrary subgroup of . In fact, we are about to see the property's rarity for a subgroup.

The symmetries of two-faced equilateral -gons, called dihedral group , is collection of all rigid body operations applicable to a equilateral -gon: imagine there's an equilateral -gons piece and a slot matching its shape, you are allowed to take out the piece and do some operations to it, and them place it back to the slot. The piece is made of some really hard material unknown to human that you can't twist, dice or change its shape in any way you know.

Under such constraint, it won't be hard for you to figure out you can't do anything with the piece except for rotating and flipping it. Since it's two faced, let's label each vertex of the piece with counter-clockwisely for the front face, and clockwisely for the back face.

Since the slot is immobile, our rotation is constrained to radian. In fact we don't even bother considering how many degrees we should rotate, but difference in edge index after rotation. When the front face is up and it is atomic when we rotate it clockwisely so that each vertex's index is offseted by modulo-. Let's denote such rotation as , and all other rotation with front face up can be viewed as repeating such process times, which can be denoted as , and obviously .

What if flipping is enrolled? It won't be hard to find that flipping the shape once and then rotating it is equivalent to rotating the shape in some way and then flipping it once. More precisely, let's denote the flipping operation as , and it won't be hard to verify .

And it won't be hard to verify that all symmetries of the equilateral -gon is made up of rotations and rotations follwed by flipping . There compositions are closed and satisfying group axioms, and thus is generalized as dihedral group .

All front face operations form a subgroup of . There're cosets and associated with the subgroup. Consider the operations on cosets, there're since , since , since , and since . So there's quotient group associated with .

The identity and flipping operations also form a subgroup of . There're cosets associated with the subgroup. However when we take two cosets and out and multiply them, there'll be , , , and . Since all cosets have elements and the product of two cosets has elements, it is not closed and there'll be no quotient group associated with .

The case study of provides us with some useful examples about conditions for quotient group to exist. Some might suspect that must be commutative (as is commutative) for quotient group to exist but judging from the non-commutative subgroup this condition is too strong. And in the same group () it is possible that some subgroup has quotient group associated with it () while some hasn't (). So the existence of quotient group completely depends on the choice of subgroup.

In fact, we can do some extra exploration on the subgroups and : every left coset of equals to the right coset, while doesn't (by ). And verifying whether all left cosets are equal to right cosets is a much easier than taking arbitrary elements from two cosets and verify them. So is it possible that is a sufficient and necessary condition for ?

Its sufficiency can be easily verified: , there'll be . And since , .

To prove its necessarity, noted that requires , which means . So we can easily get , which is equivalent to , or in other word.

So we've verified that is completely equivalent to . And if it is true for subgroup of , is also called a normal subgroup.

When subgroup of is normal, the quotient group exists. We also denote such relation between normal subgroup and group as .

It won't be hard to verify that for any group , , and are called the trivial normal subgroups of . A group without non-trivial normal subgroup is called a simple group.

When a group is not simple, we might study the behaviour of its normal subgroup and quotient group, and combine them together to describe the behaviour of the original group.

When a group is finite, we can repeat the process of breaking down until it stops at some finite normal subgroup that is simple.

Conclusion

In this text, we kindly introduce the concept of group, and the most basic but powerful tool for studying groups: by studying their substructures.

When there's a subgroup inside a group, it is only possible for the subgroup to partition the group into inter-bijective disjoint cosets, or it will violate one or more of the group or subgroup's properties.

An important special case of a subgroup is being a normal subgroup, in which all its left cosets are equal to its right cosets. In this case, not only the group properties are preserved in the normal subgroup, but all cosets also form a group called quotient group, exhibiting congruence-like behaviour.

Mastering the concept and language of group and subgroup is crucial and indispensible when one would like to dig into group theory.