What is the least positive integer $n > 1$ such that $x^n$ and $x$ are conjugate, for every $ x\in S_{11}$

Question

What is the least positive integer $n > 1$ such that $x^n$ and $x$ are conjugate, for every $ x\in S_{11}?$

Choose the correct option

$a. 10$

$b. 11$

$c. 12$

$d.13$

My attempt :Two elements $x, y \in S_n$ are conjugate if and only if they have the same cycle type.

Now take $y=x^n \implies x$ and $y $ are conjugate if gcd($|x|,n$)$=1$

If we take $|x|=11$ then least positive integer $n $ is $10$

I have no idea how to solve this problem

Answer 1

Based on your attempt, you can observe that you need the value of $n$ to be relatively prime to every possible cycle length you can construct in $S_{11}$.