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Degree of irreducible representation over reals divides twice the group order

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Statement

Suppose G is a finite group and d is the degree of an irreducible linear representation of G over the field of real numbers \R. Then, d divides 2|G|, where |G| denotes the order of G.

Facts used

  1. Degree of irreducible representation divides group order (where the irreducible representation is over an algebraically closed field of characteristic zero)