For $n\ge 5,\forall e\neq\rho\in A_n,\exists\sigma=\alpha\rho\alpha^{-1}$ for some $α\in A_n$ s.t. $ρ(i)=σ(i)$ for some $i\in\{1,2,...,n\}$, but $ρ≠σ$

Question

For $n\ge 5$, for all $e\neq \rho\in A_n, \exists \sigma=\alpha\rho\alpha^{-1}$ for some $\alpha\in A_n$ such that $\rho(i)=\sigma(i)$ for some $i\in\{1,2,...,n\}$, but $\rho\neq\sigma$.

The solution given is as follows:

Let $(123...r)$ be the largest cycle in the disjoint cycle structure of $\sigma$. Let $r\ge 3$. Then $\sigma=(123456...r)\tau$. Let $ρ=(345)\sigma(354)=(124536...r)\alpha $ for some $\alpha $. Then $\sigma(1)=2=\rho(1)$. But $\sigma(2)=3\neq 4=\rho(2)$. So $\sigma\neq\rho$.

Let $r=2$. Then $\sigma$ is a product of disjoint transpositions. Let there be at least $3$ transpositions. Then $\sigma=(12)(34)(56)\tau$. Let $\rho=(12)(35)\sigma(35)(12)=(12)(54)(36)\alpha$ for some $\alpha$. Then $\sigma(1)=2=\rho(1)$ and $\sigma(3)=4\neq 6=\rho(3)$. So $\sigma \neq \rho$.

Finally, let $\sigma=(12) (34)$. Then $\rho=(132) (12)(34)(123)=(13)(24)$. So $\sigma(5)=5=\rho(5)$. But $\sigma\neq\rho$, as required.

However, I dont get how are they concluding "Then $\sigma=(123456...r)\tau$. Let $\rho=(345)\sigma(354)=(124536...r)\alpha $ for some $\alpha $". What is $\tau$ here? Also, how did they perform the calculation: $$\rho=(345)\sigma(354)=(124536...r)\alpha.$$

EDIT 2: After the benignant clarification by Hawaiian Earring Group

I am confused about the solution for it says, "Let $(123...r)$ be the largest cycle in the disjoint cycle structure of $\sigma$", while there might not be any such cycle of $\sigma$. Again, I dont seem to understand the part following it i.e "Let $r=2$. Then $\sigma$ is a product of disjoint transpositions. Let there be at least $3$ transpositions. Then $\sigma=(12)(34)(56)\tau$. Let $\rho=(12)(35)\sigma(35)(12)=(12)(54)(36)\alpha$ for some $\alpha$." First of all, how does they assert $\sigma $ is a product of disjoint transpositions. If for a moment, we still consider $\sigma $ as a product of disjoint transpositions, but then what do they mean by "Let there be at least $3$ transpositions. Then $\sigma=(12)(34)(56)\tau$."? Next they say, "Let $\rho=(12)(35)\sigma(35)(12)=(12)(54)(36)\alpha$ for some $\alpha$." Didn't we just consider $\alpha$ as a variable previously to define $\sigma$ ? Then it does not make any sense to me.

Now, I am doubting the validity of the solution. Is it all valid? If not, can anyone suggest me a solution of this question.

I am not quite getting it...

Answer 1

$\tau $ is just the permutation $(123456\dots r)^{-1}\sigma .$

The second answer is that conjugation preserves cycle structure. And we have, on an individual cycle, $a=(a_1a_2\dots a_s)$, that the conjugate by $b\in S_n$ is

$\boxed{bab^{-1}=(b(a_1)b(a_2)\dots b(a_s))}.$

So $$\rho:=(345)\sigma (354)=(345)(123456\dots r)\tau(354)=(345)(123456\dots r)(354)(345)\tau (354)=(124536\dots r)\alpha, $$ where $\alpha: =(345)\tau (354).$ This is because $(345)$ does the following: $3\to4,4\to5,5\to3.$