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.

slides | list of my articles on automorphism towers | abstract at Bristol

${\rm P}^f\neq {\rm NP}^f$ for almost all $f$

  • J. D. Hamkins and P. D. Welch, “${\rm P}^f\neq {\rm NP}^f$ for almost all $f$,” MLQ Math.~Log.~Q., vol. 49, iss. 5, pp. 536-540, 2003.  
    @ARTICLE{HamkinsWelch2003:PfneqNPf,
    AUTHOR = {Hamkins, Joel David and Welch, Philip D.},
    TITLE = {{${\rm P}^f\neq {\rm NP}^f$} for almost all {$f$}},
    JOURNAL = {MLQ Math.~Log.~Q.},
    FJOURNAL = {MLQ.~Mathematical Logic Quarterly},
    VOLUME = {49},
    YEAR = {2003},
    NUMBER = {5},
    PAGES = {536--540},
    ISSN = {0942-5616},
    MRCLASS = {03D65 (03D10 03E45 68Q05 68Q15)},
    MRNUMBER = {1998405 (2004m:03163)},
    MRREVIEWER = {Peter G.~Hinman},
    DOI = {10.1002/malq.200310057},
    URL = {http://dx.doi.org/10.1002/malq.200310057},
    eprint = {math/0212046},
    archivePrefix = {arXiv},
    primaryClass = {math.LO},
    }

Abstract. We discuss the question of Ralf-Dieter Schindler whether for infinite time Turing machines $P^f = NP^f$ can be true for any function $f$ from the reals into $\omega_1$. We show that “almost everywhere” the answer is negative.