# Sh:882

• Kaplan, I., & Shelah, S. (2009). The automorphism tower of a centerless group without choice. Arch. Math. Logic, 48(8), 799–815.
• Abstract:
For a centerless group G, we can define its automorphism tower. We define G^{\alpha}: G^0=G, G^{\alpha+1}= Aut(G^\alpha) and for limit ordinals G^\delta= \bigcup_{\alpha<\delta}G^\alpha. Let \tau_G be the ordinal when the sequence stabilizes. Thomas’ celebrated theorem says \tau_G< 2^{|G|})^{+} and more. If we consider Thomas’ proof too set theoretical, we have here a shorter proof with little set theory. However, set theoretically we get a parallel theorem without the axiom of choice. We attach to every element in G^\alpha, the \alpha-th member of the automorphism tower of G, a unique quantifier free type over G (whish is a set of words from G* \langle x\rangle). This situation is generalized by defining “(G,A) is a special pair”.
• Version 2009-07-19_11 (24p) published version (17p)
Bib entry
@article{Sh:882,
author = {Kaplan, Itay and Shelah, Saharon},
title = {{The automorphism tower of a centerless group without choice}},
journal = {Arch. Math. Logic},
fjournal = {Archive for Mathematical Logic},
volume = {48},
number = {8},
year = {2009},
pages = {799--815},
issn = {0933-5846},
mrnumber = {2563819},
mrclass = {03E25 (20A10 20F28)},
doi = {10.1007/s00153-009-0154-2},
note = {\href{https://arxiv.org/abs/math/0606216}{arXiv: math/0606216}},
arxiv_number = {math/0606216}
}