# Sh:882

- Kaplan, I., & Shelah, S. (2009).
*The automorphism tower of a centerless group without choice*. Arch. Math. Logic,**48**(8), 799–815. arXiv: math/0606216 DOI: 10.1007/s00153-009-0154-2 MR: 2563819 -
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} }