Centralizer

For a group G and subset S, the centralizer C_G(S) is the subgroup of elements of G that commute with every element of S.

Group: A set equipped with an associative binary operation, an identity element, and inverses for every element. A centralizer is defined inside a group and inherits its operation.
Center of a group: The subgroup of a group consisting of elements that commute with every element of the group. It is the centralizer obtained when the specified subset is the entire group.
Conjugacy class: The set of elements obtained by conjugating a fixed group element by every group element. An element's centralizer is the stabilizer governing the size of its conjugacy class.
Fixed-point subgroup: The subgroup of elements fixed by every automorphism in a specified family. Centralization can be expressed as being fixed by the inner automorphisms induced by the specified elements.
Subgroup: A subset of a group that is itself a group under the same operation. The elements commuting with a fixed subset always form a subgroup.
Centralizer of a subgroup: The subgroup of elements commuting with every element of a given subgroup. This common specialization applies the definition to a subgroup rather than an arbitrary subset.
Orbit-stabilizer theorem: A theorem equating an orbit's size with the acting group's size divided by its stabilizer's size. Conjugation turns the centralizer into the stabilizer in this counting formula.
Commuting graph: A graph whose vertices are group elements, with edges joining distinct commuting elements. A centralizer is the neighborhood, together with the element itself, around a specified set in this graph.
Commutator: An expression measuring the failure of two group elements to commute, commonly defined as x⁻¹y⁻¹xy. An element centralizes S exactly when its commutators with all members of S are identity.
Normalizer: The subgroup of elements whose conjugation sends a given subgroup to itself. A normalizer preserves the subgroup as a whole; a centralizer fixes each of its elements.

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