Strong approximation methods in group theory LMS/EPSRC Short course

Here ZS = Z[1/p | p ∈ S]. In case (b) we can deduce from the Strong Approximation Theorem that ∆1 has many finite images, in particular the groups ∏k i=1 G(Fpi) whenever p1, . . . , pk are distinct primes outside S. Now, for all but finitely many primes p the group G(Fp) is semisimple, in fact it is a perfect central extension of a product of simple groups (of fixed Lie type over Fp). The simple groups of Lie type are very well understood and this enables us to deduce properties of the profinite completion ∆̂ of ∆.