Subgroup

A subset of a group that contains the identity and is closed under the group operation and inverses. With the inherited operation, it is itself a group.

Group (mathematics): A set equipped with an associative binary operation, an identity element, and inverses for every element. A subgroup inherits its operation and identity from this larger structure.
Subgroup test: A criterion that determines whether a nonempty subset of a group is a subgroup by checking closure conditions. It replaces checking every group axiom separately with a compact test.
Stabilizer (group theory): The subgroup of group elements that leave a specified object fixed under a group action. Symmetries fixing an object form a subgroup automatically.
Submonoid: A subset of a monoid that contains its identity and is closed under its associative operation. It need not contain inverses, unlike a subgroup.
Binary operation: A rule that combines each ordered pair of elements of a set to produce another element of that set. Closure requires the group's operation to keep pairs of subgroup elements inside the subset.
Generated subgroup: The smallest subgroup containing a specified set of elements. It constructs subgroups by closing chosen elements under products and inverses.
Isotropy subgroup: The subgroup of a group action that fixes a particular point. It describes the residual symmetry at one point of a space.
Subsemigroup: A subset of a semigroup closed under its associative binary operation. It may lack an identity and inverses required of a subgroup.
Identity element: An element that leaves every element unchanged when combined with it under a binary operation. Every subgroup must contain the identity of the ambient group.
Coset: A translate of a subgroup by a group element, formed by multiplying the subgroup on one side or the other. Cosets partition a group according to a chosen subgroup.

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