# What happens when one iteratively computes the automorphism group of a group? Temple University, Philadelphia 2012

This is a talk I shall give for the Mathematics Colloquium at Temple University, April 23, 2012.

The automorphism tower of a group is obtained by computing its automorphism group, the automorphism group of that group, and so on, iterating transfinitely. The question, known as the automorphism tower problem, is whether the tower ever terminates, whether there is eventually a fixed point, a group that is isomorphic to its automorphism group by the natural map. Wielandt (1939) proved the classical result that the automorphism tower of any finite centerless group terminates in finitely many steps. This was successively generalized to larger and larger collections of groups until Thomas (1985) proved that every centerless group has a terminating automorphism tower.  Building on this, I proved (1997) that every group has a terminating automorphism tower.  After giving an account of this theorem, I will give an overview of work with Simon Thomas and newer work with Gunter Fuchs and work of Philipp Lücke, which reveal a set-theoretic essence for the automorphism tower of a group: the very same group can have wildly different towers in different models of set theory.

# The automorphism tower problem for groups, Bristol 2012

Isaac Newton 20th Anniversary Lecture.  This is a talk I shall give at the University of Bristol, School of Mathematics, April 17, 2012, at the invitation of Philip Welch.

The automorphism tower of a group is obtained by computing its automorphism group, the automorphism group of that group, and so on, iterating transfinitely. The question, known as the automorphism tower problem, is whether the tower ever terminates, whether there is eventually a fixed point, a group that is isomorphic to its automorphism group by the natural map. Wielandt (1939) proved the classical result that the automorphism tower of any finite centerless group terminates in finitely many steps. This was successively generalized to larger and larger collections of groups until Thomas (1985) proved that every centerless group has a terminating automorphism tower.  Building on this, I proved (1997) that every group has a terminating automorphism tower.  After giving an account of this theorem, I will give an overview of my work with Simon Thomas, as well as newer work with Gunter Fuchs and work of Philipp Lücke, which reveal a set-theoretic essence for the automorphism tower of a group: the very same group can have wildly different towers in different models of set theory.

# Degrees of rigidity for Souslin trees

• G. Fuchs and J. D. Hamkins, “Degrees of rigidity for Souslin trees,” Journal of Symbolic Logic, vol. 74, iss. 2, p. 423–454, 2009.
[Bibtex]
@ARTICLE{FuchsHamkins2009:DegreesOfRigidity,
AUTHOR = {Fuchs, Gunter and Hamkins, Joel David},
TITLE = {Degrees of rigidity for {S}ouslin trees},
JOURNAL = {Journal of Symbolic Logic},
FJOURNAL = {Journal of Symbolic Logic},
VOLUME = {74},
YEAR = {2009},
NUMBER = {2},
PAGES = {423--454},
ISSN = {0022-4812},
CODEN = {JSYLA6},
MRCLASS = {03E05},
MRNUMBER = {2518565 (2010i:03049)},
MRREVIEWER = {Stefan Geschke},
URL = {http://wp.me/p5M0LV-3A},
doi = {10.2178/jsl/1243948321},
eprint = {math/0602482},
archivePrefix = {arXiv},
primaryClass = {math.LO},
file = F,
}

We investigate various strong notions of rigidity for Souslin trees, separating them under Diamond into a hierarchy. Applying our methods to the automorphism tower problem in group theory, we show under Diamond that there is a group whose automorphism tower is highly malleable by forcing.

# Changing the heights of automorphism towers by forcing with Souslin trees over $L$

• G. Fuchs and J. D. Hamkins, “Changing the heights of automorphism towers by forcing with Souslin trees over $L$,” Journal of Symbolic Logic, vol. 73, iss. 2, p. 614–633, 2008.
[Bibtex]
@ARTICLE{FuchsHamkins2008:ChangingHeightsOverL,
AUTHOR = {Fuchs, Gunter and Hamkins, Joel David},
TITLE = {Changing the heights of automorphism towers by forcing with {S}ouslin trees over {$L$}},
JOURNAL = {Journal of Symbolic Logic},
VOLUME = {73},
YEAR = {2008},
NUMBER = {2},
PAGES = {614--633},
ISSN = {0022-4812},
CODEN = {JSYLA6},
MRCLASS = {03E35},
MRNUMBER = {2414468 (2009e:03094)},
MRREVIEWER = {Lutz Struengmann},
URL = {http://wp.me/p5M0LV-3l},
doi = {10.2178/jsl/1208359063},
eprint = {math/0702768},
archivePrefix = {arXiv},
primaryClass = {math.LO},
file = F,
}

We prove that there are groups in the constructible universe whose automorphism towers are highly malleable by forcing. This is a consequence of the fact that, under a suitable diamond hypothesis, there are sufficiently many highly rigid non-isomorphic Souslin trees whose isomorphism relation can be precisely controlled by forcing.

In an earlier paper with Simon Thomas, “Changing the heights of automorphism towers,” we had added such malleable groups by forcing, and the current paper addresses the question as to whether there are such groups already in L.

# How tall is the automorphism tower of a group?

• J. D. Hamkins, “How tall is the automorphism tower of a group?,” in Logic and Algebra, Y. Zhang, Ed., Providence, RI: AMS, 2002, vol. 302, p. 49–57.
[Bibtex]
@INCOLLECTION{Hamkins2001:HowTall?,
AUTHOR = {Hamkins, Joel David},
TITLE = {How tall is the automorphism tower of a group?},
BOOKTITLE = {{Logic and Algebra}},
SERIES = {Contemporary Math.},
VOLUME = {302},
PAGES = {49--57},
PUBLISHER = {AMS},
YEAR = {2002},
MRCLASS = {20E36 (03E35 20A15 20F28)},
MRNUMBER = {1928383 (2003g:20048)},
MRREVIEWER = {Martyn R.~Dixon},
editor = {Yi Zhang},
doi = {10.1090/conm/302},
url = {http://wp.me/s5M0LV-howtall},
}

The automorphism tower of a group is obtained by computing its automorphism group, the automorphism group of that group, and so on, iterating transfinitely by taking the natural direct limit at limit stages. The question, known as the automorphism tower problem, is whether the tower ever terminates, whether there is eventually a fixed point, a group that is isomorphic to its automorphism group by the natural map. Wielandt (1939) proved the classical result that the automorphism tower of any finite centerless group terminates in finitely many steps. This was generalized to successively larger collections of groups until Thomas (1985) proved that every centerless group has a terminating automorphism tower. Here, it is proved that every group has a terminating automorphism tower. After this, an overview is given of the author’s (1997) result with Thomas revealing the set-theoretic essence of the automorphism tower of a group: the very same group can have wildly different towers in different models of set theory.

# 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, p. 139–157, 2000.
[Bibtex]
@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 $V^{P}$.

# Every group has a terminating transfinite automorphism tower

• J. D. Hamkins, “Every group has a terminating transfinite automorphism tower,” Proc.~Amer.~Math.~Soc., vol. 126, iss. 11, p. 3223–3226, 1998.
[Bibtex]
@article {Hamkins98:EveryGroup,
AUTHOR = {Hamkins, Joel David},
TITLE = {Every group has a terminating transfinite automorphism tower},
JOURNAL = {Proc.~Amer.~Math.~Soc.},
FJOURNAL = {Proceedings of the American Mathematical Society},
VOLUME = {126},
YEAR = {1998},
NUMBER = {11},
PAGES = {3223--3226},
ISSN = {0002-9939},
CODEN = {PAMYAR},
MRCLASS = {20E36 (20F28)},
MRNUMBER = {1487370 (2000e:20057)},
DOI = {10.1090/S0002-9939-98-04797-2},
URL = {http://jdh.hamkins.org/everygroup/},
eprint = {math/9808014},
archivePrefix = {arXiv},
primaryClass = {math.LO},
}

The automorphism tower of a group is obtained by computing its automorphism group, the automorphism group of that group, and so on, iterating transfinitely. Each group maps canonically into the next using inner automorphisms, and so at limit stages one can take a direct limit and continue the iteration. The tower is said to terminate if a fixed point is reached, that is, if a group is reached which is isomorphic to its automorphism group by the natural map. This occurs if a complete group is reached, one which is centerless and has only inner automorphisms. Wielandt [1939] proved the classical result that the automorphism tower of any centerless finite group terminates in finitely many steps. Rae and Roseblade [1970] proved that the automorphism tower of any centerless Cernikov group terminates in finitely many steps. Hulse [1970] proved that the the automorphism tower of any centerless polycyclic group terminates in countably many steps. Simon Thomas [1985] proved that the automorphism tower of any centerless group eventually terminates. In this paper, I remove the centerless assumption, and prove that every group has a terminating transfinite automorphism tower.