Abstract
We determine the Zariski-dense subgroups of Chevalley groups and their twisted analogues over infinite algebraic extensions of finite fields. It turns out that these are essentially forms of the same group (possibly becoming twisted) over smaller infinite fields. It follows from our classification that if\(\bar G\) is a simple algebraic group over the algebraic closure of a finite field, then a dense subgroup of\(\bar G\) can never be maximal, and so the maximal subgroups of\(\bar G\) are necessarily closed. It follows that Seitz’s determination of the closed maximal subgroups of\(\bar G\) actually gives all the maximal subgroups.
It also enables us to prove that ifG is a simple Chevalley group or twisted type over an infinite algebraic extension of a finite field, then in every non-trivial permutation representation ofG, every finite subgroup has a regular orbit. It follows that every non-trivial permutation module forG over a fieldk iskG-faithful. This is relevant to a programme of studying ideals in group rings of simple locally finite groups.
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To John Thompson in recognition of his many outstanding contributions to group theory
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Hartley, B., Zalesskii, A.E. On simple periodic linear groups— Dense subgroups, permutation representations, and induced modules. Israel J. Math. 82, 299–327 (1993). https://doi.org/10.1007/BF02808115
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DOI: https://doi.org/10.1007/BF02808115