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

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    @Sexyfunc$t$ion: Probably you should rewri$t$e the question to say '_Must_ the action have a fixed point?', since that seems to be what you're actually asking.2012-04-25

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(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 $m \in \Bbb{Z}$.

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    I made a comment that indicates that I might be the downvoter, but I am not... So, must be someone else, who did not bother to explain.2012-05-10