= Real character and real conjugacy class counting theorem
For a <finite group>, the number of <irreducible characters> which are <real-valued characters> equals the number of <real conjugacy classes>. On the <complex vector space> of <class functions>, the <linear operator> $Tf(g)=f(g^{-1})$ permutes the <basis> of class indicators by inversion and the <basis> of <irreducible characters> by pointwise <complex conjugation>, using the <character inversion identity>. The <trace> of a <basis> permutation is its number of fixed members. Equality of the <trace> in the two <bases> proves the theorem. The <linear operator> $T$ is complex-linear; pointwise <complex conjugation> on arbitrary <class functions> is not.
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