Finite group

A finite group is a group whose underlying set contains finitely many elements. Its order is the number of those elements.

Group (mathematics): A set equipped with an associative binary operation, an identity element, and an inverse for every element. Finite groups are groups whose underlying sets happen to be finite.
Lagrange's theorem: A theorem stating that the order of a subgroup of a finite group divides the group’s order. It is the basic divisibility constraint on finite subgroup structure.
Symmetric group: The group of all permutations of a finite set, with composition as its operation. It provides a concrete model of finite groups through permutations.
Classification of finite simple groups: A mathematical classification describing all finite simple groups in a finite list of families and exceptional cases. It organizes the simple building blocks from which finite groups can be analyzed.
Group order: The number of elements in a finite group. It measures the size that distinguishes one finite group from another.
Cauchy's theorem (group theory): A theorem stating that a prime dividing a finite group's order is the order of some element in the group. It guarantees elements of prime order from divisibility alone.
Dihedral group: The symmetry group of a regular polygon, consisting of rotations and reflections. Its geometric symmetries give a standard noncommutative family.
Representation theory of finite groups: The study of finite groups through their linear actions on vector spaces. Representations translate group structure into matrices and linear algebra.
Subgroup: A subset of a group that is itself a group under the inherited operation. Subgroups reveal how a finite group decomposes into smaller groups.
Sylow theorems: Theorems describing the number, existence, and conjugacy of subgroups whose orders are maximal powers of a prime. They extract powerful structural information from prime factors of the group order.

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