Special pairs and automorphisms of centreless groups

Lücke Philipp

Abstract

Let G be a group, A be a subset of the domain of G and L_A be the first-order language of group theory expanded by constant symbols for elements in A. We call the pair special if every element g of G is uniquely determined by the set qft_{G,A}(g) consisting of all L_A-terms t(v) with one free variable and t^G(g) = 1_G. The pair is strongly special if qft_{G,A}(g) \subseteq qft_{G,A}(h) implies g=h for all g,h in G. Special pairs were introduced by Itay Kaplan and Saharon Shelah to analyze automorphism towers of centreless groups. The purpose of this note is the further analysis of special pairs and their interaction with automorphism towers. This analysis will allow us to prove an absoluteness result for the first three stages of the automorphism tower of countable, centreless groups. Moreover, we develop methods that enable us to construct a variety of examples of such pairs, including special pairs that are not strongly special.