A question about the fixed point and group action. A group $G$ of order $35$ act on a set $S$ that has $16$ elements. Must the action have a fixed point? Why?

(Essentially Jason's argument.)
Yes, every action of this group should have a fixed point. 
Size of orbits divide the order of the group (comes from Orbit-Stabilizer Lemma). So, your orbits should be of size $1$, $5$, $7$ or $35$. 
Now, since the set is of cardianlity $16$, we cannot have an orbit of size 35. Now, suppose  there we no fixed points, then: 

for $n, m \ge 1$, $7n+5m=16$ has integer solutions in $n$ and $m$.  

Now one should argue this is not the case. 

 For $n \ge 2$, $16=7n+5m \ge 14+5m$ which would mean, $5m \le 2$ which is impossible with $m \in \Bbb{Z}$. So, $n=1$. But, then this means, $5m=11$ which is again absurd as $m \in \Bbb{Z}$. 

