The automorphism tower of a centerless group (mostly) without choice
Logic
2007-05-23 v1 Group Theory
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*< x>). This situation is generalized by defining ``(G,A) is a special pair''.
Cite
@article{arxiv.math/0606216,
title = {The automorphism tower of a centerless group (mostly) without choice},
author = {Itay Kaplan and Saharon Shelah},
journal= {arXiv preprint arXiv:math/0606216},
year = {2007}
}