Past exam of the mathematics course of the University of Oxford / 2015 / Prelims / Mathematics I / 7 / b / i / Solution

The orbits of a group action form a partition of a set $S$: every element lies in its own orbit of a group action, and two intersecting orbits of a group action coincide because each point can be moved back and then forward using inverse elements. By the orbit-stabilizer theorem, each orbit of a group action has cardinality dividing $8$, so its size is $1,2,4$ or $8$.

If every orbit of a group action had size $2,4$ or $8$, their sizes would add to an even number, contrary to $|S|=17$. Therefore there is a one-element orbit of a group action, say $\{s^{\prime }\}$. Every $g\in G$ sends $s^{\prime }$ into this same singleton, soThus $s^{\prime }$ is a fixed point of a group action. In fact the number of fixed points of a group action is odd, since all other orbits of a group action have even cardinality.