What’s the relation between the number of conjugacy classes in a finite group and the number of irreducible representations?
Solution
Assume representations are finite-dimensional over \(\CC\). Let \(k(G)\) denote the number of conjugacy classes of the finite group \(G\).
The space \(\operatorname{Cl}(G)\) of complex-valued class functions on \(G\) has dimension \(k(G)\): a class function is determined independently by one value on each conjugacy class.
On the other hand, the irreducible characters of \(G\) form an orthonormal basis of \(\operatorname{Cl}(G)\) for the inner product \[ \langle\chi,\psi\rangle =\frac1{|G|}\sum_{g\in G}\chi(g)\overline{\psi(g)}. \] Orthogonality gives linear independence. Completeness follows from the regular representation, or equivalently from the standard character-theoretic fact that every class function is a linear combination of irreducible characters.
Therefore the number of irreducible complex characters, equivalently the number of isomorphism classes of irreducible complex representations, equals the dimension of \(\operatorname{Cl}(G)\): \[ \boxed{\#\operatorname{Irr}(G)=k(G).} \] Thus a finite group has exactly as many irreducible complex representations as conjugacy classes.