Question: Show that if g is a k-cycle, then $\displaystyle fgf^{-1}$ is also a k-cycle, for any permutation $\displaystyle f$.

If the cycle decomposition of either $\displaystyle f$ or $\displaystyle f^{-1}$ was disjoint with g then they would commute and I'd be left with just $\displaystyle g$ which is a k-cycle but I'm not sure that $\displaystyle g$ is disjoint to either the cycle decomposition of $\displaystyle f$ or $\displaystyle f^{-1}$. What am I missing?

Thanks!