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Let $G: = S_7$ and $U: = \{(17), (1273)\}$. What is the order of $U$? Is $\{1, 7\}$ an Orbit from $U$?

Attempt:

I know/see $(17)$ has the order $2$ and $(1273)$ the order $4$, but I don't know how to show the order for the subgroup.

Orbit: $1^U = \{1,7\}$ (with $1^{id}= 1, 1^{(1273)}= 2, 1^{(1372)} = 3, 1^{(17)}= 7$), so no orbit?

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    $\begingroup$ If you intend $U$ to be a subgroup (rather than just the two listed elements, the appropriate notation is $\langle (1~7), (1~2~7~3) \rangle$. You're correct that the $U$-orbit that includes $1$ also includes $2$ and $3$, as well as $7$. $\endgroup$ Mar 30, 2021 at 16:38

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