Conjugacy

Conjugacy is a relation between elements in a group (and more generally in rings and algebras) defined by the condition that two elements a and b are conjugate if there exists an element g with gag^{-1} = b. It is an equivalence relation: every element is conjugate to itself, and conjugacy is preserved under the group operation. The set of all elements conjugate to a given element a is called the conjugacy class of a.

A key related concept is the centralizer C_G(a), the set of elements that commute with a. The size of the conjugacy class of a is the index [G : C_G(a)]. Consequently, in finite groups, the group is partitioned into disjoint conjugacy classes.

Examples help illustrate the idea. In the symmetric group S_n, two permutations are conjugate if and only if they have the same cycle type. In matrix groups, conjugate elements are called similar: A and B are conjugate if there exists an invertible P with A = PBP^{-1}. Over algebraically closed fields, similarity is analyzed via invariants like the Jordan form.

Conjugation also defines a natural group action. Conjugation by g gives an inner automorphism Ad_g(x) = gxg^{-1}, and the set of all such automorphisms forms the inner automorphism group Inn(G), which is isomorphic to G/Z(G), where Z(G) is the center.

Beyond abstract algebra, the concept appears in dynamical systems: two systems f and g are conjugate if a homeomorphism h satisfies h∘f = g∘h, indicating a form of dynamical equivalence. In representation theory, characters are class functions, meaning they are constant on conjugacy classes.