Changing the heights of automorphism towers

If G is a centreless group, then tau(G) denotes the height of the automorph ism tower of G. We prove that it is consistent that for every cardinal lamb da and every ordinal alpha < lambda, there exists a centreless group G such that (a) tau(G) = alpha; and (b) if p is any ordinal such that 1 less than or equal to beta < lambda, th en there exists a notion of forcing P, which preserves cofinalities and car dinalities, such that tau(G) = beta in the corresponding generic extension V-P. (C) 2000 Elsevier Science B.V. All rights reserved.