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: Amer.~Math.~Soc., 2002, vol. 302, pp. 49-57.  
    @INCOLLECTION{Hamkins2001:HowTall?,
    AUTHOR = {Hamkins, Joel David},
    TITLE = {How tall is the automorphism tower of a group?},
    BOOKTITLE = {Logic and algebra},
    SERIES = {Contemp.~Math.},
    VOLUME = {302},
    PAGES = {49--57},
    PUBLISHER = {Amer.~Math.~Soc.},
    ADDRESS = {Providence, RI},
    YEAR = {2002},
    MRCLASS = {20E36 (03E35 20A15 20F28)},
    MRNUMBER = {1928383 (2003g:20048)},
    MRREVIEWER = {Martyn R.~Dixon},
    editor = {Yi Zhang},
    doi = {http://dx.doi.org/10.1090/conm/302},
    }

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.

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