Group Theory

This post will assume some fundamental knowledge of Set Theory, as well as Functional Analysis. Only the basic ideas, e.g. what a set is, what functions are, what a bijection is, and so on are required.

Ideally, I should have titled this post "Algebra", however I hesitate to do this, as there are many misconceptions about what algebra actually is. In middle-school people learn it informally as equations with letters replacing some of the numbers. In fact, algebra is the study of relations between mathematical objects.

A group G is a set which has a built-in binary function, F(x, y). This function takes two elements from the set and returns one element from the set. This property is known as closure and is one of the requirements for a group. The other properties a group must have, with respect to its binary function are:

1) An identity element. This is an element e for which F(e, x) = F(x, e) = x. Think of this as the number 0, when we add integers. 0 + x = x + 0 = x, for any integer x. Note that the identity element is unique.

2) Inverses. Every element of the set must also have an inverse in the set. Two elements a and b are inverses of each other if F(a, b) = F(b, a) = e, where e is the identity. This is similar to the negative of a number. When adding integers, x + (-x) = (-x) + x = 0. Note that each element has a unique inverse. This includes the identity, which is its own inverse.

3) Associativity. Using the addition example, this means that (a + b) + c = a + (b + c). That is, the order in which the additions are carried out does not matter; whereas the order of the elements added does.

The last part may seem strange, in light of the addition example. After all, 2 + 3 = 3 + 2 = 5. The order is irrelevant! This is, in fact, not the case, with more general groups. For example, if we take the set of 2x2 matrices with non-zero determinant, under normal matrix multiplication, they form a group. But in this group AB is not equal to BA. The order of the elements added (or multiplied) matters in many groups. The groups for which a + b = b + a are known as abelian groups; the integers under addition are such a group.

Having read all these rather abstract notions which define a group, one may ask what the point of Group Theory is. Why should it be a field of study, when it seems so disconnected from anything which can relate to the experience of our senses? In fact, the same question could be asked of most high-level fields of Mathematics, and the answer will roughly be the same. There are in fact three reasons to study Group Theory, or indeed any sort of Mathematics. The first is because of the applications they have. With Group Theory these may not be readily apparent, however modern Physics, Chemistry and Cryptography rely on the study of Group Theory. The second reason is that we humans have a natural curiosity and a tendency to look for patterns. In many ways, the study of Mathematics in general and Algebra in particular is the study of patterns. The third reason is that mathematical study helps one develop a rational way of thinking. Abstracting problems into mathematical models is the only way which has been discovered so far, which is consistently successful in solving said problems.