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> Suppose that $G$ is a finite group, acting via homeomorphisms on $B^n$, the closed $n$-dimensional ball. Does $G$ have a fixed point?
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A *fixed point* for $G$ is a point $p \in B^n$ where for all $g \in G$ we have $g\cdot p = p$.
Notice that the answer is "yes" if $G$ is cyclic, by the Brouwer fixed point theorem. Notice that the answer is "not necessarily" if $G$ is infinite. If it helps, in my application I have that the action is piecewise linear. 


First I thought this was obvious, then I googled around, then I read about Smith theory, and now I'm posting here. 



Do finite groups acting on a ball have a fixed point?
The answer is **no**.


A fixed point free action of the finite group $A\_5$ on a $n$-cell was constructed by Floyd and Richardson in their paper [An action of a finite group on an n-cell without stationary points](https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-65/issue-2/An-action-of-a-finite-group-on-an-n-cell/bams/1183523039.full), *Bull. Amer. Math. Soc.* Volume **65**, Number 2 (1959), 73-76.


For some non-existence results, you can see the paper by Parris [Finite groups without fixed-point-free actions on a disk](https://projecteuclid.org/journals/michigan-mathematical-journal/volume-20/issue-4/Finite-groups-without-fixed-point-free-actions-on-a-disk/10.1307/mmj/1029001152.full), *Michigan Math. J.* Volume **20**, Issue 4 (1974), 349-351.


