# How many $g \in S_6$ are there such that $gxg^{-1} =x?$

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.

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