# 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.