Assume we have two affine rotations of the plane around two different fixed points. Do they generate an infinite group?
2026-04-06 21:01:34.1775509294
Do any two affine rotations with no common fixed point generate an infinite group?
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If you have a group $G$ of affine self-transformations of a vector space $V$: then if $G$ is finite then $G$ fixes $\frac1{|G|}\sum_{g\in G}g(0)$.
By contraposition, if $G$ is generated by a subset $S$ and there is no common fixed point for elements of $S$, then $G$ is infinite.