Why is group theory important

In particular, we show several methods to make new groups from known groups by considering subgroups and quotient groups. Then, we consider to classify known groups by using the concept of group isomorphism. In Section 4, we discuss and give many examples of finite groups, including symmetric groups, alternating groups, and dihedral groups. Then we give the classification theorem for finite abelian groups, which we can regard as an expansion of the Chinese remainder theorem.

In Section 5, we consider to classify elements of groups by the conjugation and discuss the decomposition of a group into its conjugacy classes. In Section 6, we explain basic facts in representation theory of finite groups. In particular, we review representations of groups, irreducible representations, and characters. Finally, we give several examples of character tables of well-known finite groups.

In Section 8, we consider finite-oriented graphs and their automorphisms. The automorphism group of a graph strongly reflects the symmetries of the graph. We remark that the reader can read this section without the knowledge of the facts in Sections 5 and 6.

In this section, we fix some notation and conventions and review some definitions in the set theory and the linear algebra:. In other words, the bijective map is one-to-one correspondence between X and Y.

There are hundreds of textbooks for the group theory. Venture to say, if the reader wants to learn more from a viewpoint of symmetries, it seems to be better to see [ 2 ]. For high motivated readers, see [ 3 , 4 ] for mathematical details. Let G be a set. We call the element e the unit of G. According to the mathematical convention, we write 1 G or simply 1 , for the unit. The product is a binary operator on G and is also called the multiplication of G.

In general, if the product of a group G is additive, then G is called an additive group. We remark that N is not a group with the usual addition since any element does not have its inverse. We remark that R with the usual multiplication is not a group since 0 does not have its inverse. In general, if the product of a group G is multiplicative, then G is called a multiplicative group. Then U n with the usual multiplication of C forms a group.

Geometrically, U n is the set of vertices of the regular n -gon on the unit circle in the complex plane C. The sixth roots of unity. In general, for a group G , if G consists of finitely many elements, then G is called a finite group. If G is not a finite group, then G is called an infinite group.

The group U n is a finite group of order n , and the groups discussed in E1 and E2 are infinite groups. Furthermore, we denote by GL 2 K the set of elements of M 2 K whose determinant is not equal to zero:. Then M 2 K with the usual addition of matrices forms an additive group. On the other hand, the set GL 2 K with the usual multiplication of matrices forms a multiplicative group. The group GL 2 K is called the general linear group of degree 2. But the most significant difference between them is the commutativity of the products.

Since group theory is an abstract itself, it had better for beginners to have sufficiently enough examples to understand it. In order to make further examples, we consider several methods to make new groups from known groups. The first one is a subgroup. Let G be a group. We can consider H itself is a group by restricting the product of G to H.

For any group G , the one point subset 1 G is a subgroup of G. We call this subgroup the trivial subgroup of G. Let us consider some other examples:. E6 Consider the group U 6 consisting of 6th power roots of unity. Then we can consider U 2 and U 3 are subgroups of U 6. The subset. We call SL 2 K the special linear group of degree 2.

In general, we can construct a subgroup from a subset of a group. Let S be a subset of a group G. Then the subset. We call S the subgroup of G generated by S. The elements of S are called generators of the subgroup S.

Here we give some examples:. E8 The additive group Z is generated by 1. In general, a group generated by a single element is called a cyclic group. Thus, Z is an infinite cyclic group, and U n is a finite cyclic group.

Next, we consider a relation between the orders of a finite group and its subgroup. Let G be a group and H a subgroup of G. Then, since the product of Z is written additively, a left coset of n Z is given by. Hence all left cosets of n Z in Z are given by. Then we can see that. Hence there exist three left cosets of U 2.

In example E11 , we can see that the order of U 2 times the number of left cosets of U 2 is equal to six, which is the order of U 6. This is no coincidence. Then we have the following: Theorem 3. As a corollary, we obtain the following: Corollary 3. If G is a finite group of prime order , then G is a cyclic group.

For a group G and its subgroup H , the set of left cosets of H is denoted by. In general, this set does not have a natural group structure.

Here we consider a condition to make it a group. Let N be a subgroup of G. If G is abelian group, any subgroup of G is a normal subgroup. As mentioned above, we have many examples of groups. Here, we consider relations between groups and examine which ones are essentially of the same type of groups. To say more technically, we classify groups by using isomorphisms.

Let G and H be groups. Namely, an isomorphism is a map such that it is one-to-one correspondence between the groups and that it preserves the products of the groups. It is, however, not an isomorphism since f is not injective. Then f is an isomorphism. Indeed f is injective. Hence f is surjective. Moreover, we have. Then f is an isomorphism since f is bijective, and. Let G and H be isomorphic groups. Then, even if G and H are different as a set, they have the same structure as a group. This means that if one is abelian, finite or finitely generated, then so is the other, respectively.

In other words, for example, an abelian group is never isomorphic to a non-abelian group and so on. Remark that this is not a matrix. Let S n be the set of permutations on X. Then the set S n with this product forms a group. We call it the symmetric group of degree n. The symmetric group S n is a finite group of order n! Since S 1 is the trivial group, and.

Hence, S 3 is non-abelian. Here we consider another description of permutations. We call a cyclic permutation of length 2 a transposition. Namely, any transposition is of type. A cyclic permutation of length 1 is nothing but the identity permutation:. In general, a permutation cannot be written as a single cyclic permutation but a product of some cyclic permutations which do not have a common letter. For example, consider. Remark that two cyclic permutations which do not have a common letter are commutative.

The importance of group theory was emphasized very recently when some physicists using group theory predicted the existence of a particle that had never been observed before, and described the properties it should have. Later experiments proved that this particle really exists and has those properties. The group theory is the most crucial ingredient in the present day of science, mathematics, statistics and computer science.

It was ascertained in the nineteenth century in association with delivering solutions for algebraic expressions. In particular, the group was the set of all the permutations of the roots of an algebraic expression that exhibits the characteristics that the combination of any two of these permutations belongs to the set.

And later on, the belief was made generalized to the notion of an abstract group. However, an abstract group is the study of a set, with an operation defined on it. There are four primary sources in the development of group theory, that are in the terms of author names and time of origin :.

Also read: What is knowledge graph? The theory of group is essentially the study of groups where a group is a set equipped with specific binary operations. For example, the set of all the integers with addition. If there are a finite number of components, the group is termed as a finite group, where the number of components is called the group order of the group.

A subset of the group which is bounded under any of the group operations is known as Subgroups. The group theory is the branch of abstract-algebra that is incurred for studying and manipulating abstract concepts involving symmetry.

It is the tool which is used to determine the symmetry. Also, symmetry operations and symmetry components are two fundamental and influential concepts in group theory. In core words, group theory is the study of symmetry, therefore while dealing with the object that exhibits symmetry or appears symmetric, group theory can be used for analysis. Must catch: 7 branches of Discrete Mathematics?

The simple answer is that Group Theory is the systematic and methodical study of symmetry, for an instance, when a physical quantity possess some sort of symmetry and so it undergoes any kind of symmetry operation in order to obtain a simplified description through accounting the outcomes of that symmetry. Outcomes, obtained from group theory, could only be useful if and only if an individual understands them well enough to consider and deploy them up and provide users with a few basics insights as earlier as possible.

Group theory appears in the situation where the symmetry plays a significant part and its roots fall in Galois' study of the symmetries of the roots of polynomials.

This group abstractly measures the "symmetries" of that object, under the sort of transformations we're considering important. Symmetry has shown itself to be a powerful tool for reasoning in many circumstances -- it allows us to take an argument about one part of a structure and carry it throughout the structure to many other places where it applies. Because essentially every object we study in mathematics has an automorphism group, it makes sense to study groups in general, so that we don't have to start afresh in each branch of mathematics when we want to consider symmetries of whatever it is that we're studying.

To answer your question "Why should be bother about pure math" and as a hindsight, just focuss on the application stuff , I always look at the results that were purely abstract w decade ago, but do pop up widely in consumer electronics like Goppa- codes in CDROMs and DVD. Also, I once answered at my aunts reaction: "But can you buy something at the grocery with it?

Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams? Learn more. Why should we care about groups at all? Ask Question. Asked 10 years, 9 months ago. Active 10 years, 2 months ago. Viewed 10k times. So here's what I'm looking for: Are there any problems that that 1 don't originate from group theory, 2 have very elegant solutions in the framework of group theory, and 3 are completely intractable or at the very least, extremely cumbersome without non-trivial knowledge of groups?

Elliott Elliott 3, 5 5 gold badges 23 23 silver badges 47 47 bronze badges. In particular, by treating the different permutations of the cube as a finite group and then using Lagrange's Theorem, the maximum number of moves needed to solve the cube was determined. I'll admit this may not be of any practical value, but such information would be hard to find using other methods. The value of Group Theory is in it's generality.

The definition of a group is so simple that many real-world problems can give rise to a group, and so much is known about the structure of groups. Caring about exact solutions to polynomial equations is fairly concrete algebra.

But that is their practical use of them. How ever their structure on their own is worth study for those who think group structures are beautiful on their own. Show 2 more comments. Active Oldest Votes. Community Bot 1. Alex B. Since your profile says that any feedback is welcome, here is some feedback: it's usually safer to ask questions when you don't understand something, than throwing confident statements out there and distributing downvotes.

You might also like to check peoples' profiles before correcting them on what is or isn't a group. My list of examples was intended as purely illustrative, it is not a formal introduction to group theory, and there is no need to introduce heavy notation.

In the over 4. I can live with your downvote, I was merely satisfying your request for feedback. While I was cycling home, I thought of an example to illustrate this abuse of notation: strictly speaking, it is wrong to speak of "the ring of integers", since the integers are merely a set.

But every mathematician would prefer to take the minute risk of ambiguity to saying "the ring whose underlying set is the set of integers, and where the structure operations are addition with 0 as the neutral element, and multiplication with 1 as the neutral element".

Life is just too short. Show 5 more comments. Qiaochu Yuan Qiaochu Yuan k 41 41 gold badges silver badges bronze badges. I think it's safe to argue that people provably care about particle physics.