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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.
