This is the $2^{nd}$ part of a two-part question. The first part asks us to prove that the conjugate of a $j$-cycle $\sigma = (k_1 … k_j)$ by any other permutation $\tau \in S_n$ is a $j$-cycle and has the form $\tau \sigma \tau^{-1} = (\tau(k_1) … \tau(k_j))$. To which I have provided my proof here: Prove that the conjugate of a $j$-cycle is a $j$-cycle and is given by the following formula.
The $2^{nd}$ part of this question has the following problem statement:
Further, prove that if $\sigma' \in S_n$ is any other $j$-cycle, then $\sigma$ and $\sigma'$ are conjugate. Hint: You should explicitly find a conjugating element $\tau \in S_n$
Prelude: Definition of $2$ elements being conjugates
Now, for this problem, we haven't come across the exact definition of $2$ elements being "conjugates" only the concepts of
- "Conjugation of $H$ by $g$": $gHg^{-1}$ where $H < G$ and $g \in G$ and
- "Conjugate of $h$ by $g$": $ghg^{-1}$ where $h \in H$ so that conjugation of $H$ by $g$ is the set of all conjugates of $h$ by $g$
- "Conjugation by $g$": $c_g: G \to G$ defined by $c_g(h) = ghg^{-1}$ Where $c_g$ is an automorphism
I'm assuming that saying that $2$ elements are conjugates would mean that we're saying that for some $g \in G$ $$ghg^{-1} = gh'g^{-1}\ \text{for any}\ h,h' \in H$$
So, in the context of our question $\sigma, \sigma'$ being conjugates means that for some $\tau \in S_n$ we have $$\tau \sigma \tau^{-1} = \tau \sigma' \tau^{-1}\ \text{for any}\ \sigma, \sigma' \in S_n$$
Is this correct?
The body of the problem
Let $\sigma' = (l_1 … l_j) \in S_n$ be a $j$-cycle. Note $\sigma \neq \sigma'$. Then since $\tau \in S_n$ a lemma in the book guarantees the existence of an automorphism $c_{\tau}: S_n \to S_n$ defined by $$c_{\tau}(\sigma') = \tau \sigma' \tau^{-1} = (\tau(l_1) … \tau(l_j))$$ Where $\tau \sigma' \tau^{-1}$ is a $j$-cycle, by part $1$.
Now, I'm assuming we want to find $\tau \in S_n$ such that the above definition of $\sigma,\sigma'$ being conjugates is satisfied. I'm just having some trouble seeing how to show that these $2$ objects are equivalent. Any guidance or suggestions are very much appreciated!
Best Answer
Your interpretation of elements being conjugate is not quite right. We say that $h$ and $h'$ are conjugate if there exists a $g$ such that $$ h' = ghg^{-1}. $$ Equivalently, $h'=c_g(h)$ in your notation. This may not look symmetric, but note that $h'=c_g(h) \iff h=c_{g^{-1}}(h')$. Now, note that we only do the conjugation on one side of the equation. In fact, if we took your interpretation, we would have $$ ghg^{-1} = gh'g^{-1} \iff h=h', $$ so that's clearly a bad definition of anything.
With that out of the way, you need to find (construct) a permutation $\tau$ such that $\tau\sigma'\tau^{-1} = \sigma$. You already wrote that $$ c_\tau(\sigma') = (\tau(l_1) \cdots \tau(l_j)), $$ so how do you define $\tau$ so that this cycle equals $\sigma$? You're allllmost there.