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In the $S_7$ group, in action on itself by $xy=xyx^{-1}$, I would like to calculate the orbit of $(123)(456)$.

I read the definition:

Consider a group $G$ acting on a set $X$. The orbit of an element $x$ in $X$ is the set of elements in $X$ to which $x$ can be moved by the elements of $G$. The orbit of $x$ is denoted by $G\cdot x$: $G\cdot x = \{g\cdot x | g\in G\}$

But I can't seem to understand how to calculate technically speaking the orbit of $(123)(456)$. My goal is to calculate $|Stab_{S_7}(123)(456)|$. In order to achieve it I have to calculate the orbit, but how?

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    $\begingroup$ What action are you studying? Presumably $S_7$ acting on itself, but by conjugation, or one of the regular actions? $\endgroup$ Jan 20, 2019 at 12:52
  • $\begingroup$ @user3482749 Sorry, I meant on itself, ill edit. $\endgroup$
    – vesii
    Jan 20, 2019 at 12:57
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    $\begingroup$ There is more than one action of $S_7$ on itself. Do you mean the action by conjugation? $\endgroup$ Jan 20, 2019 at 13:00
  • $\begingroup$ @user3482749 on itself by $xy=xyx^{-1}$ $\endgroup$
    – vesii
    Jan 20, 2019 at 13:01

1 Answer 1

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If you have a permutation written down as its cycle decomposition, it's very easy to describe the action of an element by conjugation - that element is applied to all of the entries in the cycles. If $y$ includes the cycle $(a_1a_2\dots a_k)$, $\sigma y\sigma^{-1}$ includes the cycle $(\sigma(a_1)\sigma(a_2)\dots\sigma(a_k))$.

So then, we start with an element that's a composition of two disjoint $3$-cycles. Conjugate by something, and we get a composition of two disjoint $3$-cycles. Can you see what the orbit will be now?

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  • $\begingroup$ Ahh.. Still can see it, I fully understand the theorems, but I can't seem to understand it when technically speaking. Can you please show me one solved example (the one I asked or some other example) so I would know how to solve it formally? $\endgroup$
    – vesii
    Jan 20, 2019 at 13:17

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