Abstract Algebra

Introduction

In mathematics, we often encounter objects that are sets with various operations that can be performed on them. For instance, one may add and multiply integers, and one can do the same with rational numbers, real numbers, and even complex (or imaginary) numbers. Or, given two functions that input and output real numbers, we can compose them. We can add vectors, or multiply them by scalars.

In abstract algebra, we attempt to provide lists of properties that common mathematical objects satisfy. Given such a list of properties, we impose them as “axioms”, and we study the properties of objects that satisfy these axioms. The objects that we deal with most in the first part of these notes are called groups, rings, and fields.

Groups, rings, and fields all sets with binary operations. A binary operation inputs two elements of the set and outputs a third such element. Addition and multiplication of integers, for instance, are binary operations, as is composition of real-valued functions of a real number. Scalar multiplication of a vector in the plane is not however, since it starts not with two vectors, but rather a scalar (i.e., a real number) and a vector.

We often require our binary operations to have certain properties like associativity or commutativity. If we call our operation “$\star$”, then associativity reads

and commutativity reads

In imprecise terms, they tell you that the order in which you perform the operations doesn’t matter. Though the most typically-encountered binary operations tend to be associative, many are not commutative (i.e., are “noncommutative”). For instance, you may recall that the order of composition of functions matters: e.g., $\sin (x^2)$ and ${\sin }^2(x)$ are two different things.

The integers $\mathbb{Z}$, the rational numbers $\mathbb{Q}$, the real numbers $ℝ$, and the complex numbers $ℂ$ are all rings, as is the set of $n$ by $n$ matrices with entries in any of these. A ring is a set with two binary operations called addition and multiplication. In order to be a ring, we require associativity of both operations, commutativity of addition, and distributivity of the two operations. Distributivity can be expressed as follows:

Every ring must have an element called “$0$” (satisfying $0+x=x$ for any $x$) and, for every number $x$, there should be another $-x$ which when added to $x$, gives you $0$. Typically, a ring also has an element called “$1$”, which satisfies $1\cdot x=x=x\cdot 1$. For instance, in any ring of $n$ by $n$ matrices, the element “$1$” is actually the identity matrix.

You may recall that multiplication of square matrices is noncommutative (if they are at least $2$ by $2$ in size). For instance, we have

On the other hand, the multiplications in $\mathbb{Z}$, $\mathbb{Q}$, $ℝ$, and $ℂ$ are all commutative. A ring in which multiplication is a commutative binary operation is a called a commutative ring.

Once we have rings, fields are simple to describe. Fields are commutative rings with one extra property. That is, a field has inverses under multiplication: if $x$ is in the field and isn’t $0$, then there must be an element $x^{-1}=1∕x$ as well, and it satisfies $x\cdot x^{-1}=1$. In particular, $\mathbb{Q}$, $ℝ$, and $ℂ$ are fields as well as rings, but $\mathbb{Z}$ is not a field. In a field, fractions add and multiply in the familiar way:

Some rings have nonzero elements $x$ and $y$ with product $\mathit{𝑥𝑦}$ equal to $0$. These are called zero-divisors. For instance,

and so we can have that the product of two nonzero matrices is the zero matrix. If a commutative ring has no zero divisors, then we can construct its field of fractions artificially. Its elements consisting of elements denoted $x∕y$, where $x$ and $y$ are in the original rings. The field of fractions of $\mathbb{Z}$ is $\mathbb{Q}$, and here we have our first example of a construction that is well-known for the simplest ring of all, the integers, but can be performed more generally (for instance to polynomials), starting from the axioms of a ring and a few extra properties.

Groups may seem a bit less familiar, but they are also in a sense simpler. Groups have only one binary operation. Call it whatever you like: addition, multiplication, or just “$\star$”. A group and its binary operation $\star$ must satisfy just three properties: associativity of $\star$, the existence of an identity element $e$, and the existence of inverses. The identity element $e$ is like the number $1$ is under multiplication, or like $0$ is under addition, in the rings that are familiar to us. It satisfies

for all $x$ in the group. The inverse of an element $x$ is normally denoted $x^{-1}$, but it is written $-x$ if our operation is addition. It satisfies

In particular, rings are groups if we forget about the multiplication and just consider the operation of addition. Fields are groups under multiplication if we throw out $0$.

Many less familiar but interesting mathematical objects are groups. The rotations of a circle form a group under composition (following one rotation by another), and the permutations (switches of positions) of five balls between five slots are a group under composition as well. The $n$ by $n$ real matrices with nonzero determinant form a group under multiplication too. The set of “moves” of a Rubik’s cube (compositions of rotations of sides by 90 degree multiples) form a group too: a very complicated one, in fact. So, groups are in some sense a less refined but much broader class of objects than the rings, with more exotic members.

In our examples, some of the groups have finitely many elements and hence are known as finite groups. Here’s an interesting property of every finite group. Suppose that a finite group $G$ has $n$ elements, and let $x$ be one of them. Then $x^n$, which is $x\star x\star \cdots \star x$ with $x$ appearing $n$ times, is the identity element $e$. For instance, if I permute the position of $5$ balls in five slots in a certain manner, over and over, the balls will wind up in the position they started after $120$ steps, since that is the order of the group. In fact, this exaggerates the number of repetitions needed: the balls end up at the starting point in six or fewer. The same goes with the Rubik’s cube: repeat the same sequence of moves enough times, and, if you have enough patience (meaning watch out for carpal tunnel syndrome), you will end back up where you started. This is something that, a priori, may not seem obvious at all. Yet, this property of finite groups is a very general phenomenon, derived solely from the group axioms.

Hopefully this encourages you to believe that abstract algebra may be of serious use both inside and outside mathematics, and indeed, it is so, in addition to being a fascinating and beautiful theory in its own right for those so inclined. In the next chapter, we begin our study of abstract algebra at a much more leisurely pace.