Let the group $G \subset S_7$, $|G|=8$. Show that there exists $i\in{1,...7}$ such that for all $f\in G,f(i)=i$.
I have attempted this problem, but I am not entirely sure if my thinking is correct. Using orbit stabilizer theorem, I write $|G|=|orb_G(i)||stab_G(i)|$, and since $|G|=8, $ then $|stab_G(i)|$ is equal to either $1,2,4,8$ and $|orb_G(i)|$ equal to $8,4,2,1$ respectively. Now, when the order of the orbit is 8 and stabilizer 1, can I conclude that there exists a function that maps $i$ to $i$?
Any help would be appreciated!