Some remarks on unipotent automorphisms

An automorphism α of the group G is said to be n-unipotent if [g,_n α] = 1 for all g ∈ G. In this paper we obtain some results related to nilpotency of groups of n-unipotent automorphisms of solvable groups. We also show that, assuming the truth of a conjecture about the representation theory of solvable groups raised by P. Neumann, it is possible to produce, for a suitable prime p, an example of a f.g. solvable group possessing a group of p-unipotent automorphisms which is isomorphic to an infinite Burnside group. Conversely we show that, if there exists a f.g. solvable group G with a non nilpotent p-group H of n-automorphisms, then there is such a counterexample where n is a prime power and H has finite exponent.