A group $G$ of order $35$ act on a set $S$ that has $16$ elements. Must the action have a fixed point? Why?
A question about the fixed point and group action.
2
$\begingroup$
group-theory
finite-groups
-
1@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
1 Answers
2
(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 $m \in \Bbb{Z}$.
-
0I 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