ON SIMPLE PERIODIC LINEAR-GROUPS - DENSE SUBGROUPS, PERMUTATION REPRESENTATIONS, AND INDUCED MODULES

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 (pos sibly becoming twisted) over smaller infinite fields. It follows from our classification that (G) over bar is a simple algebraic group over the algebraic closure of a finite field, then a dense subgroup of (G) over bar can never be maximal, and so the maximal subgroups of (G) ove r bar are necessarily closed. It follows that Seitz's determination of the closed maximal subgroups of (G) over bar actually gives all the m aximal subgroups. It also enables us to prove that if G is a simple Ch evalley group or twisted type over an infinite algebraic extension of a finite field, then in every non-trivial permutation representation o f G, every finite subgroup has a regular orbit. It follows that every non-trivial permutation module for G over a field k is kG-faithful. Th is is relevant to a programme of studying ideals in group rings of sim ple locally finite groups.