Degree of irreducible representation divides index of center

From Groupprops
Jump to: navigation, search

This article gives the statement, and possibly proof, of a constraint on numerical invariants that can be associated with a finite group

This article states a result of the form that one natural number divides another. Specifically, the (degree of a linear representation) of a/an/the (irreducible linear representation) divides the (index of a subgroup) of a/an/the (center).
View other divisor relations |View congruence conditions

Statement

Let G be a finite group and k be an algebraically closed field of characteristic zero. Let \rho be an irreducible representation of G over k. Then, the degree of \rho divides the index [G:Z(G)] of the center Z(G) in G. (Note that since the quotient by the center equals the inner automorphism group, this is equivalent to saying that the degree of \rho divides the order of the inner automorphism group).

Related facts

Similar facts about degrees of irreducible representations

Analogous facts about size of conjugacy class