A Note on Automorphisms of Free Nilpotent Groups

We exhibit normal subgroups of a free nilpotent group F of rank two and class three, which have isomorphic finite quotients but are not conjugate under any automorphism of F . A remarkable fact about free profinite groups of finite rank is that any isomorphism between finite quotients of such a group F lifts to an automorphism of F . This is true, more generally, if F a free pro-Cgroup of finite rank, where C is a family of finite groups closed under taking subgroups, homomorphic images and direct products, and containing nontrivial groups. A proof and the relevant definitions can be found in [FJ86, Proposition 15.31], but the essence of the argument goes back to Gaschütz [Gas55]. In preparation for a summer school on “Zeta functions of groups” held by Marcus du Sautoy and the author in June 2002 in Trento (Italy), du Sautoy suggested that this may remain true for (abstract) free nilpotent groups F , and asked the author, who was responsible for that part of the course, to provide a proof. If confirmed, this claim would have simplified the course by avoiding the need to set up the language of profinite groups. Unfortunately, this claim already fails for F a free abelian group of rank one, that is, an infinite cyclic group: in this case F has exactly two automorphism, while its quotient of order n has φ(n) automorphisms, and φ(n) > 2 for n > 4. A milder statement which would have been sufficient for our purposes would be that any two normal subgroups of F with finite isomorphic quotients are conjugate under some automorphism of F . This is also false, and one does not have to dig much deeper in order to find a counterexample. We first record an example suggested by the anonymous referee. It is based on a three-generated group of order p and class two, which was studied in [DH75]. After that we present an example where F is two-generated and the quotients have order p. This order is easily seen to be minimal for such an example. 2000 Mathematics Subject Classification. Primary 20E05; secondary 20F18.