group theory homework solutions

group theory homework solutions

Group Theory Homework Solutions

1. Introduction

Group theory is a mathematical theory that is used to describe the symmetry in a variety of different contexts. At its core, group theory is the study of the different ways in which individual elements can be combined to make up a larger structure. The main goal of this first chapter is to introduce you to the world of group theory and to get you started on the course. It’s important to realize that this is quite an abstract area of mathematics, and so it’s not always immediately obvious what group theory is actually for. One of the main aims of this course is to demonstrate that group theory is an incredibly powerful tool, and that it has a number of vital applications in several different areas of mathematics, and in the sciences as a whole. However, we will also see that the subject is interesting and important in its own right, and no prior knowledge of where group theory ‘comes in useful’ should be assumed. The first thing we need to do is to define what a group actually is. Be prepared for an abstract definition. A group is a set, G, together with a single binary operation, that satisfies four very important properties: that the group is closed under the operation, that the operation is associative, that there is an identity element in the group, and that every element has an inverse. This might all sound a bit technical and challenging, but it’s easier to understand than it might first seem. We will break down each one of these properties in turn over the next couple of chapters and you will start to become accustomed to understanding and manipulating the definition. Also, importantly, these four properties mean that a group on its own is a very general kind of structure, and in some sense is not particularly restrictive. However, what makes group theory such a powerful subject is that many familiar mathematical systems turn out to be groups under certain operations, and the properties of groups impose very strong and restrictive conditions. This enables us to prove that some systems are essentially uniquely defined by the given operation and properties associated with it, and it also allows for the efficient and exhaustive classification of mathematical objects by understanding what kind of groups they belong to.

2. Basic Concepts

Basic Concepts section focuses on introducing the fundamental concepts and principles of group theory. At the outset, we will define what is a group. Formally, a group is a set G with a binary operation on G such that the following 3 conditions are satisfied. First, for any two elements a and b in G, their product a * b is also in G. Second, the product should be associative, i.e. for any three elements a, b, and c in G, (a * b) * c = a * (b * c). And finally, there exists an element e in G, such that for every element a in G, the product e * a equals a * e equals a. We call e the identity element of the group. If there exists an element h in G, such that the product h * a equals e, where e is the identity element, we call h the inverse of the element a. The order of a group is the number of elements in the group. If the order of a group is finite, we say that the group is a finite group; if the order of a group is infinite, we say that the group is an infinite group. One important point is that the product of any two elements in a group should also be in the group. This property is known as the closure property and it will be used in many proofs.

3. Problem Solving Techniques

There are no clear-cut methods for solving problems in group theory. Like most mathematics, problem solving is a creative as well as systematic. It is imperative to be familiar with the usual notation and terminology of group theory, to be able to use and follow precisely the statements of the theorems and to be able to recognise what theorems might be applicable in a given situation. These points cannot be overemphasised. However, one of the main skills in mathematics in general and in group theory in particular is to reduce problems to an appropriate level of simplicity unmatched which the key ideas may begin to emerge. Often, this will involve looking for patterns of some sort and this is where the ‘creative’ part of problem solving comes into play. For the student new to group theory, what group to consider and what element to consider as the starting point may not be at all obvious and indeed might follow only after a number of failed attempts. Equally, the theorems that might lead out of the given information and so begin the reduction of the problem to a simpler form may not all be at once clear. Regular practice and determined effort are the only ways to develop intuition into the subject and to learn what questions to ask when faced with a new problem.

4. Advanced Topics

By virtue of looking at the order of the elements, it appears that the group G is cyclic with a generator. Just as I have found out. In fact, by Lagrange’s theorem, mainly because the order of the group and the order of the cyclic group is the school of. On which this proof could have been based, which turns out to be useful in other directions. That is, the result that the kernel of a group homomorphism is a normal subgroup of the domain group is, in fact, the foundation of an elementary, but as yet unattained, result on the solvability of polynomial equations. So this exercise ties together two different facets of group theory. On the one hand, it ties to the very fundamental study of the cycle structure of any particular permutation. On the other hand, it indicates the utility of results concerning the mapping of groups under group homomorphisms – that is, to study both sources for group and structure, and the way those structures can be mapped to one another in manners which respect that structure. And here we see that the kernel, which is defined with reference to an image, and vice versa gives rise to a normal subgroup, which can wield effectively an entire kernel in the fashion we have indicated. So, in the examination of homomorphisms and their kernel/image relations, this result will form a potent tool in the solving of other problems and establish the properties of other groups. I found that exercise very enlightening; it’s always a stimulus to release connections, especially ones hitherto not revealed. It makes group theory one of the most “naturally” wide-reaching in terms of its application, as in every source of group, there must be studied their isomorphism and homomorphism to other groups – an inspiring observation. Also, theorem 2 gives us that both the kernel of and the kernel of have no common terms, mainly because the order of the group G and the order of the group formed by are such that the total order of is. And certainly, the generator is a cyclic group of order that has no. However, by the cyclic subgroup theorem, we are guaranteed, when is a subgroup of order, that with any nonzero element in generates. Ergo, in the case at hand, generates itself. So we realize that the condition “generator” refers to the idea that every element of a cyclic group where appears must satisfy, usually, an infinite amount property. And this infinite amount of properties associated could have been used in refutation of our conclusion that. This illustrates a nice case in which a result of group theory could have been utilized as a negative point in an attempt to gain a contrasting proof property. Interesting group, I think.

5. Conclusion

It is worth mentioning that Theorem 3.72 is and must be one of our major tools in the study of finite cyclic groups, yet it is also one of the results which contributes most to the proof of the aforementioned fundamental theorem. There is a lot of interesting material for group theory one can look into, as mentioned in the introduction. Despite the fact that it is a branch of mathematics, group theory is very much self-contained. However, at the same time, it has extensive applications to various different areas of maths, e.g. number theory, geometry, ring and field theories, as well as modern physics and cryptography. Mathematically, it’s definitely worthwhile to take further study into group theory. It is hoped that this information in the notes can give some insights to beginners in this area. Thank you for reading.

We can classify the theories covering Lagrange’s theorem: the order of a subgroup divides the order of the group. We can then examine the order of any element of a finite group G by looking at the order of the cyclic subgroup generated by this element using Proposition 3.31. This result is undoubtedly useful, but more importantly, we can now explain many of the extremely important properties that Cauchy’s theorem provides. This will be summarized as Proposition 3.59. Also, by taking advice from the notes, we can check some special cases of Cayley’s theorem in Proposition 3.48. And as a follow-up, when we consider the number of generators of a finite cyclic group, Theorem 3.72 provides a direct result implying the fundamental theorem of finite cyclic groups, as opposed to Lagrange’s theorem which is relatively more general in nature.

So far, so good. We have seen considerable examples of groups throughout the notes. We have seen three different approaches in building examples, namely algebraic examples in Section 3.1, geometric examples and the notion of symmetries in Section 3.1, 3.2 and 3.3, and direct products of groups in Section 3.4 and e.g. Proposition 3.64.