Show for $\alpha\in S_n$, $\alpha^{-1}(123)\alpha=(xyz)$ for some $x,y,z$. How are $x,y,z$ related to $\alpha$? Generalize.

Question

I found a exercise in my textbook, but I could not come to the conclusion.The question says that demonstrate that for any $\alpha \in S_n$, there holds ${\alpha}^{-1}(123)\alpha=(xyz)$ with suitable $x,y,z$. How are $x,y,z$ related to $\alpha$? Genaralize your conclusion ${\alpha}^{-1}(123)\alpha$.

My attempt: I wrote some $\alpha$s to see the behavior of ${\alpha}^{-1}(123)\alpha$, but i cannot see any results. For example:

• If $\alpha=(12)$, then ${\alpha}^{-1}(123)\alpha=(132)$.

• If $\alpha=(123)$ ,then ${\alpha}^{-1}(123)\alpha=(123)$.

• If $\alpha=(12)(34)$ ,then ${\alpha}^{-1}(123)\alpha=(142)$

Can you help me to see how ${\alpha}^{-1}(123)\alpha$ is related to $\alpha$ and $(123)$?

Answer 1

Sure.

What you will find is that, for any $\alpha$,

$$\alpha^{-1}(123)\alpha=(\alpha^{-1}(1)\alpha^{-1}(2)\alpha^{-1}(3)),$$

where $\alpha^{-1}(i)$ is $\alpha^{-1}$ evaluated at $i$.

Indeed, conjugation of a permutation by another permutation preserves cycle structure. This has been covered many times before on this site.

Answer 2

The relation is $\alpha^{-1}(xyz)\alpha=(\alpha^{-1}(x)\alpha^{-1} (y)\alpha^{-1} (z)) $.

This generalizes to $n $-cycles, and leads easily to the fact that conjugation preserves cycle structure.