Algebraic structures: groups, rings and fields

Summary

This lesson, the last of the unit on matrices and algebraic structures, gathers into a few definitions what numbers, matrices, symmetries and the remainders of division have in common. It defines binary operations and their properties, and with them groups: the integers under addition, the nonzero complex numbers, the roots of unity, the rotations of the plane, the invertible matrices, the six symmetries of the equilateral triangle and the remainders of division by $n$ . It proves, without using commutativity, the uniqueness of the identity element and of the inverse, cancellation and the rule for the inverse of a product; it studies subgroups and the order of an element, and proves Lagrange’s theorem by means of cosets. It then defines rings and fields, proves that the remainders of division by $n$ form a field exactly when $n$ is prime, and presents isomorphisms as the precise way of saying that two structures are the same. It ends with a map of the two Algebra and Geometry courses and of what comes after them.

Learning objectives

Recognize binary operations and their properties, decide whether a set with an operation is a group, and build the table of a finite group.
Prove the elementary properties of groups, the uniqueness of the identity element and of inverses, cancellation and the inverse of a product, without using commutativity.
Compute the order of an element, find subgroups and apply Lagrange’s theorem to finite groups.
Distinguish rings and fields, operate with the remainders of division by $n$ , solve equations in them and decide whether two structures are isomorphic.

Binary operations

Throughout this course and the course Algebra and Geometry I, natural numbers, integers, rational, real and complex numbers have been added and multiplied; functions have been composed, matrices have been multiplied and rotations of the plane have been performed one after another. Each of these operations takes two objects of a set and returns a third object of the same set, and in all of them similar properties have been proved again and again. This lesson isolates those properties and studies what follows from them, whatever the nature of the objects.

A binary operation on a set $A$ is a rule that assigns to each ordered pair of elements $a$ and $b$ of $A$ a unique element of $A$ , written $a \cdot b$ , or simply $ab$ . That the result belongs to $A$ is expressed by saying that $A$ is closed under the operation.

The dot does not necessarily denote a product of numbers here: it is the generic name of the operation, which in each case will be addition, multiplication, composition or any other. When the operation is an addition, the sign $+$ is used by custom. Four properties are of interest. The operation is associative if $(ab)c = a(bc)$ for all $a$ , $b$ and $c$ in $A$ , and commutative if $ab = ba$ for all $a$ and $b$ . An element $e$ of $A$ is an identity element if $ea = ae = a$ for every $a$ ; and, if there is an identity element, an element $b$ is an inverse of $a$ when $ab = ba = e$ .

(ab)c = a(bc), \qquad ab = ba, \qquad ea = ae = a, \qquad ab = ba = e

None of the four properties is automatic. Subtraction is a binary operation on $\mathbb{Z}$ , but it is neither associative nor commutative, and it has no identity element: $0$ satisfies $a - 0 = a$ , but not $0 - a = a$ . Division is not a binary operation on $\mathbb{Q}$ , because it assigns nothing to the pair formed by $1$ and $0$ . Addition on $\mathbb{N}$ is associative and commutative, but it has no identity element, because in Course I the natural numbers begin at $1$ . Composition of functions is associative and not commutative, as was proved in the Course I lesson on function operations, and the same happens with the product of square matrices of the same order, studied in the lesson on matrices and matrix operations, whose identity element is the identity matrix $I$ .

(5 - 3) - 2 = 0 \neq 4 = 5 - (3 - 2)

In the rest of this lesson

Groups
Examples and counterexamples
The symmetries of the equilateral triangle
The remainders of division by $n$
Elementary properties of groups
Subgroups and the order of an element
Lagrange’s theorem
Rings and fields
Isomorphisms: the same structure
Solved problems
Inverses in $\mathbb{Z}_7$ and a linear equation
The groups in which each element is its own inverse
The invertible remainders of $\mathbb{Z}_8$ and the fourth roots of unity
The eight symmetries of the square
Wilson’s theorem
Proposed problems
Invertible remainders and zero divisors of $\mathbb{Z}_{12}$
The check digit of books
Fermat’s little theorem from Lagrange’s theorem
The groups of prime order
The groups of four elements
A map of the two courses and what comes after