What is the order of $U$? Is $\{1, 7\}$ an orbit from $U$? 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?
Let $U=\langle(17),(1273)\rangle$.
Since $(17)(1273)(17)=(1372)=(1273)^{-1}$ it follows that $H$ is a dihedral group $D_4$.
So $|U|=8$.
Let $U=\langle(17),(1273)\rangle$.
Since $(17)(1273)(17)=(1372)=(1273)^{-1}$ it follows that $H$ is a dihedral group $D_4$.
So $|U|=8$.