Introduction
Modern (abstract) algebra studies structures — sets equipped with operations satisfying specific rules — rather than numbers alone, revealing that very different mathematical objects (numbers, matrices, symmetries, polynomials) often obey identical underlying patterns. This chapter builds from the most general structures (sets, relations) up through increasingly specific ones (groups, rings, fields, vector spaces), each adding more rules and, correspondingly, more usable structure.
1. Sets, relations, mappings, and elementary logic
A relation on a set is a rule specifying which pairs of elements are related; an equivalence relation (reflexive, symmetric, and transitive) partitions a set into disjoint equivalence classes — a construction used repeatedly throughout algebra (for instance, defining cosets, below). A mapping (function) assigns each element of one set (the domain) to exactly one element of another (the codomain); mappings are further classified as injective (one-to-one: distinct inputs give distinct outputs), surjective (onto: every element of the codomain is reached), and bijective (both, allowing a well-defined inverse mapping to exist).
2. Binary operations and algebraic structures
A binary operation combines two elements of a set to produce a third element of the same set (closure). An operation may or may not be associative ((a∗b)∗c = a∗(b∗c)) or commutative (a∗b = b∗a); it may have an identity element (e∗a = a∗e = a for all a) and individual elements may have inverses (a∗a⁻¹ = a⁻¹∗a = e). Which of these properties a given operation satisfies determines which named algebraic structure (below) the set and operation form together.
3. Groups, subgroups, cyclic groups, cosets, and homomorphisms
A group is a set with a single binary operation that is closed, associative, has an identity element, and every element has an inverse. Groups formalise the idea of symmetry and reversible transformation, and are the single most important structure in this chapter. A subgroup is a subset that is itself a group under the same operation. A cyclic group is generated entirely by repeated application of the operation to a single element (its generator). Cosets of a subgroup H partition the whole group into equal-sized blocks (each of the form gH or Hg), and are the key tool behind Lagrange’s theorem — the order (size) of any subgroup must divide the order of the group, a powerful constraint on which subgroups can possibly exist. A homomorphism is a mapping between two groups that preserves the group operation (φ(a∗b) = φ(a)∗φ(b)); an isomorphism is a bijective homomorphism, meaning the two groups are structurally identical, differing only in the labels of their elements.
4. Rings, integral domains, and fields
A ring has two operations (typically called addition and multiplication): it forms a commutative group under addition, and multiplication is associative and distributes over addition, but multiplication need not be commutative and elements need not have multiplicative inverses. An integral domain is a commutative ring with no zero divisors (nonzero elements a, b such that ab = 0) — this property is exactly what allows the familiar cancellation law (if ab = ac and a ≠ 0, then b = c) to hold. A field additionally requires every nonzero element to have a multiplicative inverse, making both addition and multiplication fully invertible — the real numbers, rational numbers, and complex numbers are all fields, while the integers form only an integral domain (2 has no integer multiplicative inverse).
5. Vector spaces, subspaces, linear (in)dependence, basis, and dimension
A vector space over a field is a set of “vectors” closed under addition and scalar multiplication, satisfying a specific list of axioms (associativity, distributivity, existence of a zero vector, and so on) — a considerable generalisation beyond arrows in physical space, encompassing polynomials, matrices, and function spaces as valid “vectors” as long as the axioms hold. A subspace is a subset that is itself a vector space under the same operations. A set of vectors is linearly independent if no nontrivial combination of them sums to zero (equivalently, no vector in the set can be written as a combination of the others); a basis is a linearly independent set that also spans the space (every vector in the space can be written as some combination of basis vectors); the number of vectors in any basis of a given space is always the same, and is called the space’s dimension.
6. Linear transformations, matrices, determinants, eigenvalues, and eigenvectors
A linear transformation between vector spaces preserves vector addition and scalar multiplication; once a basis is chosen for each space, every linear transformation can be represented concretely by a matrix. The determinant of a square matrix is a single number encoding whether the corresponding transformation is invertible (nonzero determinant) and, geometrically, how the transformation scales volume (and whether it reverses orientation, if negative). An eigenvector of a matrix A is a nonzero vector v satisfying Av = λv for some scalar λ (its corresponding eigenvalue) — meaning the transformation only stretches or shrinks that particular vector (by factor λ) rather than rotating or otherwise redirecting it; eigenvalues and eigenvectors reveal a transformation’s most fundamental directions and scaling behaviour, and are central to countless applications, including the vibrational normal modes met in Physics Chapter III.
Key Takeaways
- Equivalence relations partition a set into equivalence classes; injective, surjective, and bijective mappings describe increasingly strict correspondence between two sets.
- Groups (one operation: closure, associativity, identity, inverses) are the central structure of this chapter; Lagrange’s theorem (subgroup order divides group order) is a key structural constraint, and homomorphisms/isomorphisms compare group structures.
- Rings, integral domains, and fields form a hierarchy of increasingly restrictive two-operation structures, distinguished by commutativity of multiplication, absence of zero divisors, and existence of multiplicative inverses.
- Vector spaces generalise “vectors” far beyond physical arrows; basis and dimension are well-defined because every basis of a given space has the same size.
- Matrices concretely represent linear transformations once a basis is fixed; determinants indicate invertibility and volume scaling, while eigenvalues/eigenvectors reveal a transformation’s fundamental scaling directions.