Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2004 / Paper 2 / 10 / ii / Solution

If $h=ag{a}^{-1}$, the representation law gives $D(h)=D(a)D(g)D(a{)}^{-1}$. Trace invariance under this similarity transformation givesThus the character of a representation is constant on every conjugacy class.