On groups of automorphisms of residually finite groups (Q1585301)

The main result of this paper is the following Theorem. Let \(G\) be a soluble-by-finite minimax group which is residually finite. Let \(\Gamma\) be a subgroup of \(\Aut(G)\) which induces a nilpotent group of automorphisms in each finite characteristic quotient of \(G\). Then \(\Gamma\) is nilpotent. Also nilpotent subgroups of \(\Aut(G)\) have bounded class. The main tool used in the proof is a result of independent interest. Theorem. For each positive integer \(n\) there is an infinite set of primes \(\pi_n\) such that nilpotent subgroups of \(\text{GL}_n(p)\) have class at most \(3n/2\) for all \(p\) in \(\pi_n\).