The automorphism tower of a centerless group (mostly) without choice

Details The automorphism tower of a centerless group (mostly) without choice The automorphism tower of a centerless group (mostly) without choice

2006-06-09

arxiv.org/abs/math/0606216v1

Mathematics

Logic

Scientific paper

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''.

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