Abstract Algebra

A group is a set G equipped with a binary operation (usually called multiplication) that satisfies four axioms: closure (the operation always produces another element of G), associativity (the order of grouping does not matter), identity (there exists an element that does nothing), and inverses (every element has an undo operation). Groups capture the essence of symmetry. The symmetries of a square form a group of eight elements (four rotations and four reflections). The integers under addition form an infinite group. Lagrange's theorem states that the size of any subgroup must divide the size of the whole group — a simple yet powerful constraint on group structure.

Abstract algebra studies STRUCTURE rather than specific numbers. A group is a set with one associative operation, identity, and inverses. Groups capture symmetry: the 8 symmetries of a square form the dihedral group D4. The integers under addition form an infinite abelian group. Lagrange's theorem: the size of any subgroup divides the group size. Rings add a second operation (multiplication) distributing over the first (addition). The integers and polynomials are rings. A field is a ring where division works: rationals, reals, complexes, and finite fields GF(p). Galois theory connects field extensions to groups, proving the impossibility of solving quintic equations by radicals — a problem that stumped mathematicians for 300 years.

A ring is a set with two operations (usually addition and multiplication) where addition forms an abelian group and multiplication is associative and distributes over addition. The integers form a ring. Polynomials over a field form a ring. A field is a ring where every non-zero element has a multiplicative inverse. The rational numbers, real numbers, and complex numbers are all fields. Finite fields (also called Galois fields) are crucial for cryptography and error-correcting codes. Galois theory connects field extensions to group theory, explaining why there is no general formula using radicals to solve quintic (fifth-degree) polynomial equations — the corresponding Galois group is not solvable.

Explore Cayley tables (group multiplication tables). See how group axioms constrain the possible patterns in these tables.