Group theory

Group theory studies sets equipped with an associative operation, an identity element, and inverses. Groups capture symmetry and provide a framework for comparing algebraic operations.

Group: A set with an associative binary operation, an identity element, and an inverse for every element. This algebraic structure is the basic object group theory studies.
Symmetry group: The group of transformations that preserve a specified object or structure. Symmetry groups are the central source of concrete examples in group theory.
Évariste Galois: A French mathematician whose work connected permutations of polynomial roots with solvability by radicals. His ideas made groups central to the theory of equations.
Galois theory: A mathematical theory linking field extensions to groups of field automorphisms. It uses groups to determine when polynomial equations can be solved by radicals.
Monoid: A set with an associative operation and identity element, but not necessarily inverses. Monoids retain two group axioms while dropping the requirement of inverses.
Group homomorphism: A map between groups that preserves their operations. Homomorphisms compare groups while preserving their algebraic structure.
Permutation group: A group whose elements are permutations of a set and whose operation is composition. Permutations realize abstract groups as rearrangements of elements.
Joseph-Louis Lagrange: An Italian-French mathematician whose work on permutations and equations anticipated group theory. His studies of permutations helped establish the subject's early foundations.
Lie group: A group that is also a smooth manifold, with smoothly varying multiplication and inversion. Lie groups model continuous symmetries in geometry and physics.
Semigroup: A set with an associative binary operation, without requiring an identity or inverses. Semigroups isolate associativity from the other group axioms.

Linked from 40 pages