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

Statement
The following are equal for a finite group $$G$$:


 * 1) The number of characters of $$G$$ taking values in $$\mathbb{R}$$ arising from irreducible representations of $$G$$ over $$\mathbb{R}$$.
 * 2) The number of characters of $$G$$ taking values in $$\mathbb{R}$$ arising from representations of $$G$$ over $$\mathbb{C}$$ such that no proper nonzero subrepresentation takes values entirely in $$\mathbb{R}$$.
 * 3) The number of equivalence classes of $$G$$ under real conjugacy. Each such class arises as the union of a conjugacy class and the conjugacy class of inverse elements.
 * 4) The number of homomorphisms from $$\mathbb{Z}$$ to $$G$$, up to equivalence of automorphisms of $$\mathbb{Z}$$ and inner automorphisms of $$G$$.

Caveats and corollaries
The number of irreducible representations over reals is not the same as the number of irreducible representations over the complex numbers that can be realized over the reals. The latter number is either smaller or equal, and it is equal when the group is an ambivalent group, which means that every element is conjugate to its inverse.

Also, although the counts in (1) and (2) are equal, it is possible for a real character to arise from an irreducible representation over the complex numbers that is not realized over the reals. However, some multiple of that representation can be realized over the reals. This explains the equality of counts in (1) and (2). The smallest multiple used is termed the Schur index.

Related facts

 * Number of irreducible representations equals number of conjugacy classes
 * Number of irreducible representations over complex numbers with real character values equals number of conjugacy classes of real elements
 * Number of orbits of irreducible representations need not equal number of orbits of conjugacy classes under automorphism group

Small finite groups
Note that

Total number of irreducible representations over complex numbers = (Number of irreducible representations over the complex numbers with real character value) + 2(Number of conjugate pairs of complex numbers with non-real character value)

and

Number of irreducible representations over reals = (Number of irreducible representations over the complex numbers with real character value) + (Number of conjugate pairs of complex numbers with non-real character value)

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 $$\mathbb{C}$$.