# Thread: any permutation times a k-cycle times the inverse is a k-cycle

1. ## any permutation times a k-cycle times the inverse is a k-cycle

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

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

Thanks!

2. Originally Posted by temp31415
Question: Show that if g is a k-cycle, then $fgf^{-1}$ is also a k-cycle, for any permutation $f$.

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

Thanks!
If $\sigma = (a_1,a_2,...,a_k)$ and $\tau$ is any permutation then $\tau \sigma \tau^{-1} = (\tau (a_1),...,\tau (a_k))$.

3. Originally Posted by ThePerfectHacker
If $\sigma = (a_1,a_2,...,a_k)$ and $\tau$ is any permutation then $\tau \sigma \tau^{-1} = (\tau (a_1),...,\tau (a_k))$.
I don't think I understand. Could you explain it in a bit more detail?