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Groupprops β

Class function



Symbol-free definition

A class function' on a group is defined as a function (to any set) that takes the same value on any two conjugate elements. Equivalently, it is a function on the group that is constant on conjugacy classes, and hence descends to a function from the set of conjugacy classes.

Definition with symbols

A class function on a group G is a function f from G to some set X such that f(g) = f(hgh^{-1}) for any g,h \in G.

Particular cases

Conjugacy classes of images are class functions

Let \rho:G \to H be a homomorphism. Then the function that sends each g to the conjugacy class of \rho(g) is a class function. This follows from the fact that if two elements in G are conjugate, their images in H are also conjugate.

Characters of linear representations are class functions

Further information: Character

For any linear representation, the character of that linear representation, viz the map that sends each group element to the trace of the corresponding linear operator, is a class function. This follows from the fact that the character depends only on the conjugacy class of the linear operator corresponding to the group element.