Past exam of the mathematics course of the University of Cambridge / 2020 / ia / Paper 2 / 8D / b / Solution

If $x$ is a fixed point of $f$, thenConversely, applying the same argument to $f={\sigma }^{-1}g\sigma$ transports every fixed point of $g$ back to one of $f$. Thus the bijection $\sigma$ restricts to a bijection between the two fixed-point sets, so conjugate functions have the same number of fixed points.