Number of irreducible representations over complex numbers with real character values equals number of conjugacy classes of real elements

Statement
Suppose $$G$$ is a finite group. Then, the following numbers are equal:


 * 1) The number of irreducible representations of $$G$$ over the complex numbers whose characters are real-valued. Note that this includes both real representations (representations realized over $$\R$$), and quaternionic representations, which are not realized over $$\mathbb{R}$$ but whose double is realized over $$\R$$ (so they have Schur index 2).
 * 2) The number of conjugacy classes in $$G$$ of real elements, i.e., elements that are conjugate to their inverses.

Related facts

 * Number of irreducible representations equals number of conjugacy classes
 * Number of irreducible representations over reals equals number of equivalence classes under real conjugacy

Facts used

 * 1) uses::Application of Brauer's permutation lemma to Galois automorphism on conjugacy classes and irreducible representations (follows in turn from uses::Brauer's permutation lemma): Suppose $$G$$ is a finite group and $$r$$ is an integer relatively prime to the order of $$G$$. Suppose $$K$$ is a field and $$L$$ is a splitting field of $$G$$ of the form $$K(\zeta)$$ where $$\zeta$$ is a primitive $$d^{th}$$ root of unity, with $$d$$ also relatively prime to $$r$$ (in fact, we can arrange $$d$$ to divide the order of $$G$$ because sufficiently large implies splitting). Suppose there is a Galois automorphism of $$L/K$$ that sends $$\zeta$$ to $$\zeta^r$$. Consider the following two permutations:


 * The permutation on the set of conjugacy classes of $$G$$, denoted $$C(G)$$, induced by the mapping $$g \mapsto g^r$$.
 * The permutation on the set of irreducible representations of $$G$$ over $$L$$, denoted $$I(G)$$, induced by the Galois automorphism of $$L$$ that sends $$\zeta$$ to $$\zeta^r$$.

Then, these two permutations have the same cycle type. In particular, they have the same number of cycles, and the same number of fixed points, as each other.

Proof
Given: A finite group $$G$$

To prove: The number of irreducible representations of $$G$$ over the real numbers equals the number of equivalence classes of elements of $$G$$ under real conjugacy.

Proof: Let $$C(G)$$ be the set of conjugacy classes of $$G$$ and $$I(G)$$ be the set of irreducible representations of $$G$$ over $$\mathbb{C}$$.