On Groups of Automorphisms of Residually Finite Groups

We show that certain properties of groups of automorphisms can be read off from the actions they induce on the finite characteristic quotients of their underlying group G. In particular, we obtain criteria for groups of automorphisms of a residually finite and (soluble minimax)-by-finite group G to be nilpotent or soluble. Moreover, we give explicit bounds on the class (the derived length, resp.) of such groups of automorphisms in terms of invariants of G. Finally, we consider similar questions when G is the free group of rank two.