Let $\sigma$ be the 5-cycle $(1\:2\;3\;4\;5)$ in $S_5$. Find the element $\tau \in S_5$ which accomplishes the specified conjugation

Question

I cant seem to find a solution to any of the parts to the following problem.

Let $\sigma$ be the 5-cycle $(1\:2\;3\;4\;5)$ in $S_5$. Find the element $\tau \in S_5$ which accomplishes the specified conjugation: $$1.\space\space\tau \sigma \tau^{-1} = \sigma^2$$ $$2.\space\space\tau \sigma \tau^{-1} = \sigma^{-1}$$ $$3. \space\space\tau \sigma \tau^{-1} = \sigma^{-2}$$

I tried to draw a pentagon and find some combination of flips and rotations, but I couldn't find any with the desired properties. How should I approach these problems. any help is appreciated.

Answer 1

Hint: For any $\tau, \pi\in S_n$ with $\pi=(p_1\dots p_{k_1})\dots (p_m\dots p_{k_m})$, we have $$\tau\pi\tau^{-1}=(\tau(p_1)\dots\tau( p_{k_1}))\dots (\tau(p_m)\dots \tau(p_{k_m})).$$

For $\sigma=(12345)$, we have $\sigma^{-1}=(54321),$ so if we want $\tau$ such that $\tau\sigma\tau^{-1}=\sigma^{-1}$, note that this would require, by the above, that $$\begin{align}\tau(1)&=5,\\ \tau(2)&=4,\\ \tau(3)&=3,\\ \tau(4)&=2,\text{ and }\\ \tau(5)&=1,\end{align}$$ whence $\tau=(15)(24).$