# Changing the heights of automorphism towers

• J. D. Hamkins and S. Thomas, “Changing the heights of automorphism towers,” Ann.~Pure Appl.~Logic, vol. 102, iss. 1-2, pp. 139-157, 2000.
@article {HamkinsThomas2000:ChangingHeights,
AUTHOR = {Hamkins, Joel David and Thomas, Simon},
TITLE = {Changing the heights of automorphism towers},
JOURNAL = {Ann.~Pure Appl.~Logic},
FJOURNAL = {Annals of Pure and Applied Logic},
VOLUME = {102},
YEAR = {2000},
NUMBER = {1-2},
PAGES = {139--157},
ISSN = {0168-0072},
CODEN = {APALD7},
MRCLASS = {20F28 (03E35 20A15)},
MRNUMBER = {1732058 (2000m:20057)},
MRREVIEWER = {Markus Junker},
DOI = {10.1016/S0168-0072(99)00039-1},
URL = {http://jdh.hamkins.org/changingheightsoverl/},
eprint = {math/9703204},
archivePrefix = {arXiv},
primaryClass = {math.LO},
}

If $G$ is a centreless group, then $\tau(G)$ denotes the height of the automorphism tower of $G$. We prove that it is consistent that for every cardinal $\lambda$ and every ordinal $\alpha < \lambda$, there exists a centreless group $G$ such that (a) $\tau(G) = \alpha$; and (b) if $\beta$ is any ordinal such that $1 \leq \beta < \lambda$, then there exists a notion of forcing $P$, which preserves cofinalities and cardinalities, such that $\tau(G) = \beta$ in the corresponding generic extension $V^{P}$.

## 2 thoughts on “Changing the heights of automorphism towers”

1. Can we know anything more about those Groups, or about Heights of Automorphism Groups of known Groups?