Group Terms

Group

A set with an operation that combines any two elements to form a third, obeying closure, associativity, identity, and invertibility.

Subgroup

A subset of a group that is itself a group under the same operation.

Abelian Group

A group in which the group operation is commutative.

Normal Subgroup

A subgroup that is invariant under conjugation by any element of the group.

Cyclic Group

A group generated by a single element.

Permutation Group

A group consisting of all permutations of a set.

Field

A set where addition, subtraction, multiplication, and division (except by zero) are defined and satisfy the properties of arithmetic.

Finite Field

A field with a finite number of elements.

Ring

A set equipped with two binary operations, addition and multiplication, where addition forms an abelian group and multiplication is associative.

Vector Space

A set of vectors where vector addition and scalar multiplication are defined.

Algebra

A vector space equipped with a bilinear product.

Homomorphism

A map between two algebraic structures that preserves their operations.

Isomorphism

A bijective homomorphism that has an inverse, indicating the two structures are structurally identical.

Automorphism

An isomorphism from an algebraic structure to itself.

Symmetry Group

The group of all symmetries of a geometric object, including rotations and reflections.

Group Action

A way a group "acts" on a set, preserving the structure of the set.

Kernel

The set of elements in the domain that map to the identity element under a homomorphism.

Coset

A set formed by multiplying a subgroup by a fixed element from the group.

Quotient Group

The group formed by the cosets of a normal subgroup.

Commutator

An element of the form ghg⁻¹h⁻¹, which measures how far two elements commute.