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The affine and projective groups are maximal. (English) Zbl 1397.20007

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 S. A. Adeleke and 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 F. Bogomolov and M. Rovinsky [Cent. Eur. J. Math. 11, No. 1, 17–26 (2013; Zbl 1277.20003)].

MSC:

20B27 Infinite automorphism groups
03C40 Interpolation, preservation, definability
20E28 Maximal subgroups
51E15 Finite affine and projective planes (geometric aspects)
51E10 Steiner systems in finite geometry
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References:

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