The affine and projective groups are maximal (Q2790655)

The infinite symmetric group \(S_\omega\) is a topological group with respect to the product topology of \({\mathbb N}^{\mathbb N}\). The authors show that the affine groups \(\text{AGL}_n \mathbb Q\) with \(2\leq n \leq \omega\) are maximal among the closed subgroups of \(S_\omega\). Their proof uses the classification of infinite 3-transitive Jordan groups due to \textit{S. A. Adeleke} and \textit{D. Macpherson} [Proc. Lond. Math. Soc. (3) 72, No. 1, 63--123 (1996; Zbl 0839.20002)]. A similar maximality result is proved for the projective groups \(\text{PGL}_n \mathbb Q\) with \(3 \leq n\leq \omega\). As the authors point out, this holds more generally for projective groups over arbitrary infinite fields by \textit{F. Bogomolov} and \textit{M. Rovinsky} [Cent. Eur. J. Math. 11, No. 1, 17--26 (2013; Zbl 1277.20003)].