Conjugacy class

The conjugacy class of an element in a group is the set of all elements obtained from it by conjugation. Conjugacy classes partition the group and encode how its elements relate under internal symmetry.

Conjugation (group theory): The operation sending a group element x to gxg⁻¹ for a fixed group element g. Repeated conjugation by all group elements generates the class.
Group (mathematics): A set equipped with an associative operation, an identity element, and inverses for every element. Conjugacy classes are defined using the group's operation and inverses.
Symmetric group: The group of all permutations of a finite set, with composition as its operation. In this group, conjugacy classes are classified by cycle type.
Class function: A function on a group that is constant on each conjugacy class. Characters are class functions, so their values need only be specified class by class.
Commutator: The element aba⁻¹b⁻¹ measuring the failure of two group elements to commute. Commutators generate a subgroup, whereas conjugacy classes collect elements related by conjugation.
Centralizer: The subgroup of elements that commute with every element of a specified subset. The centralizer of an element determines how many distinct conjugates it has.
Identity element: The element of a group that leaves every element unchanged under the group operation. It forms a conjugacy class by itself in every group.
Alternating group: The subgroup of even permutations in a symmetric group. Some symmetric-group classes split into multiple classes upon restriction.
Character theory: The study of traces of group representations and the information they encode about representations. Conjugacy classes index the rows and columns of character tables.
Similarity (linear algebra): A relation between square matrices in which one is obtained from the other by an invertible change of basis. Matrix similarity is the linear-algebra counterpart of conjugacy in general linear groups.

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