Skip to content
Axioma

Mathematics · Frontier · 35 min · Chapter 17 of 28

Abstract algebra

Groups, rings, and fields: the study of operations by the axioms they satisfy, not by what the elements happen to be.

Read first: The language of proof, Linear algebra, Number theory

The idea

Abstract algebra extracts the pattern and throws away the costume. The integers under addition, the nonzero reals under multiplication, the symmetries of a square, and the invertible matrices under multiplication all satisfy the same four group axioms: a binary operation that is associative, a two-sided identity, and a two-sided inverse for every element. You prove a theorem once, from the axioms, and it applies to all of them. Commutativity is extra. When it holds, the group is called abelian.

A subgroup is a subset that is itself a group under the same operation. Lagrange’s theorem says that if G is finite and H is a subgroup, then |H| divides |G|. The converse is false: a divisor of the order need not be the order of a subgroup, though it is for cyclic groups, and by Cauchy's theorem a prime divisor is always the order of some element. The symmetric group on 3 letters, with 6 elements, has subgroups of order 1, 2, 3, and 6, and no subgroup of... it does have those. A4, order 12, has no subgroup of order 6. That is the standard witness that the converse of Lagrange fails.

Rings add a second operation. The integers are the motivating example: addition makes an abelian group, multiplication is associative and distributes over addition. A field also requires that the nonzero elements form a group under multiplication. The rationals, the reals, the complexes, and the integers modulo a prime are fields. The integers modulo a composite number are not. A homomorphism is a map that preserves the operations. For groups, the kernel, the set of elements sent to the identity, is a normal subgroup, and the first isomorphism theorem says the image is isomorphic to the group modulo the kernel.

Hold onto this

  1. A group is a set with one associative operation, an identity, and inverses.
  2. Lagrange’s theorem: the order of a subgroup divides the order of a finite group.
  3. A ring has addition and multiplication. A field is a ring where every nonzero element has a multiplicative inverse.
  4. Homomorphisms are the structure-preserving maps, and their kernels measure how much they collapse.

Formulas

Group axioms

Associativity, identity, inverses. Closure is part of saying the operation maps the set to itself.

e
The identity element
a⁻¹
The inverse of a

Lagrange’s theorem

For a subgroup H of a finite group G. The index |G|/|H| counts the distinct cosets.

|G|
Number of elements, the order of G

Kernel of a homomorphism

Elements collapsed to the identity. Trivial kernel means the homomorphism is injective.

φ
A group homomorphism

Field

Modular arithmetic gives a field exactly for prime moduli.

Z/pZ
Residue classes modulo p

Worked examples

1. A small group

Show that the residues {1, 2, 3, 4} modulo 5 form a group under multiplication.

  1. The product of two of them is never divisible by 5, so the set is closed up to residues.
  2. Multiplication is associative, and 1 is the identity.
  3. Inverses: 1·1, 2·3 = 6 ≡ 1, 4·4 = 16 ≡ 1. Every element has an inverse.

It is a group of order 4. In fact it is cyclic, generated by 2: the powers of 2 mod 5 are 2, 4, 3, 1.

2. Lagrange in action

In a group of order 15, what are the possible orders of an element?

  1. The order of an element is the order of the cyclic subgroup it generates.
  2. That order divides 15, by Lagrange.
  3. Possible orders: 1, 3, 5, 15.

1, 3, 5, or 15. There is no element of order 2, 4, 6, 7, and so on.

Try these

Is the set of integers under subtraction a group?

Why do the residues modulo 4 fail to be a field?

Common mistakes

  • The converse of Lagrange is false. Do not assume a subgroup of every dividing order.
  • In a nonabelian group, ab = ba is a special event, not a default. The symmetries of a square do not all commute.
  • Kernel is a subgroup of the domain, not of the codomain. The image is the subgroup of the codomain.