Fixed Points of Small Permutation Groups

Question

Given a subgroup $G\leq S_p$, where $p$ is prime, $|G|<p$, can we guarantee that the action of $G$ on $\{1,\ldots,p\}$ has a fixed point?

Answer 1

For $p \leqslant 5$, every subgroup $G$ of $S_p$ with $\lvert G\rvert < p$ has a fixed point, as one can easily verify by considering all candidate subgroups.

For $p \geqslant 7$ this is not the case, there always is a $G$ of order $6$ that acts without fixed point, for example the cyclic subgroup generated by the fixed-point-free permutation $$(1,2,3)(4,5)\ldots (p-1,p)\,.$$