Degree of irreducible representation over field of characteristic coprime to order divides product of order and Euler totient function of exponent

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Suppose G is a finite group, k is a field whose characteristic is relatively prime to the order of G, and \varphi is an irreducible representation of G over k (which need not be absolutely irreducible, i.e., it may split over some algebraic extension of k). Then, the degree of \varphi divides the product of the order of G and the Euler totient function of the exponent of G.

Related facts

Other facts in general

Facts over a splitting field

We can obtain much better bounds on the degrees of irreducible representations if K is a splitting field for G. Specifically:

Facts used

  1. Sufficiently large implies splitting: Any sufficiently large field for a finite group G, i.e., a field that contains all the m^{th} roots of unity where m is the exponent of G, is also a splitting field for G.
  2. Degree of irreducible representation divides group order