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If $x = (123)(456)$ and $y = (234)(561)$, count the number of $g \in S_6$ such that

$1. gxg^{-1}=x$; and

$2. gxg^{-1}=y$

For 1, I got that $g = e_{S_6}$ (the identity in $S_6$) is one of them, and for 2, I got that $g = (123456)$ is an answer, but I have no idea how to "count" the number of $g$ in each.

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  • $\begingroup$ How about $(123)$? $\endgroup$ – Eric Towers Oct 21 '19 at 4:11
  • $\begingroup$ Yes, that works too. But my question is more geared towards how to "count" all the $g$'s that satisfy the quesiton. $\endgroup$ – CharlieCornell Oct 21 '19 at 4:12
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    $\begingroup$ What theorems are you familiar with? In particular, have you seen the Orbit-Stabilizer theorem? $\endgroup$ – HallaSurvivor Oct 21 '19 at 4:56
  • $\begingroup$ @HallaSurvivor No I am not familiar with that theorem. $\endgroup$ – CharlieCornell Oct 21 '19 at 5:04
  • $\begingroup$ Hint: The answers for part $1$ and part $2$ are equal. For part $1$, the elements $g$ that work form a subgroup isomorphic to $Z_2\times Z_3^2$. $\endgroup$ – Batominovski Oct 21 '19 at 5:05

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