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In this question, it we see that a transitive permutation group acting on a finite set with two or more elements must have a fixed-point-free element. I was wondering whether or not this result could extend to permutation groups on infinite sets.

The proofs given in the linked question rely on counting arguments so do not suggest any approach to the case for infinite groups. So, does there exist a transitive permutation group on an infinite set where every permutation has a fixed point? If so, is there some simple example? Or is this an open problem?

My first thought for finding an example was the group of homeomorphisms on a closed $n-$ball, as by Brouwer's fixed-point theorem each such homeomorphism has a fixed point. This doesn't work, however, as we cannot send a point on the boundary of the ball to a point in the interior, so the group does not act transitively.

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For a very simple example, let $S$ be any infinite set and let $G$ be the group of all finite support permutations of $S$. Then $G$ acts transitively on $S$, but every element of $G$ has a fixed point (in fact, a cofinite set of fixed points!).

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    $\begingroup$ That example is not simple, but if you restrict to even permutations (which forma normal subgroup of index $2$), of finite support then it is :) $\endgroup$
    – Derek Holt
    May 10 '21 at 7:21

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