Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · math/9808094

How Tall is the Automorphism Tower of a Group?

Joel David Hamkins
View original source ↗Read source document ↗Export metadata ↓

Abstract

The automorphism tower of a group is obtained by computing its automorphism group, the automorphism group of THAT group and so on, iterating transfinitely. Each group maps into the next using inner automorphisms and one takes a direct limit at limit stages. The question is whether the process ever terminates in a fixed point, a group which is isomorphic to its automorphism group by the natural map. In this talk I will prove that every group has a terminating automorphism tower. After this, I will discuss the set-theoretic aspects of the height of the automorphism tower of a group, and sketch my recent proof with Simon Thomas that it is consistent to have a group whose automorphism tower is wildly modified by forcing; indeed, in various models of set theory the automorphism tower of this very same group can be almost arbitrarily specified.

Explore related subjects

math.LOmath.GR

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Joel David Hamkins. 1998-08-21. How Tall is the Automorphism Tower of a Group?. https://arxiv.org/abs/math/9808094

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Natural Term Logic

In this paper we develop a formal system called Natural Term Logic (NTL). NTL aims to represent key aspects of the logical and grammatical mechanisms of natural language as well as grammatical transformations which preserve core logical meaning. NTL can be seen as a refinement of the ideas of Quine's paper `Variables Explained Away' and the technical concepts introduced by Bealer and Zalta. NTL is more fine-grained than Bealer's first-order intensional logic (BL): there is a many-to-one correspondence $ν$ between NTL terms and closed BL terms as well as a canonical map $β$ which assigns to each closed BL term a corresponding NTL term. The map $ν$ can be seen as assigning a core logical content of the NTL term. We define a series of reductions on NTL terms which intuitivelyy speaking capture meaning-preserving syntactic transformations ( transformations which preserved the basic logical meaning of a term) and our main result is that each NTL term $T$ reduces to a unique normal term $N$. The reductions fall into the structural, predicative and pushing-in categories. Predicative reductions decompose NTL terms so that predication is only applied to a primitive term (such terms are called prenormal). A key ingredient in the proof is the fact that $βνN = N$ when $N$ is normal. This suggests that within NTL the normal form of a term expresses the core logical content of the term.

math.LO↗
arXiv2026

Hyper-hyperfiniteness and complexity

We show that if there exists a countable Borel equivalence relation which is hyper-hyperfinite but not hyperfinite, then the complexity of hyperfinite countable Borel equivalence relationsis as high as possible, namely, $Σ^1_2$-complete. We also establish an implication between the question of the effectivity of hyperfiniteness and its complexity.

math.LO↗
arXiv2026

Coordinate recognition: General theory, Groups, and other surprises

A class of structures \emph{recognizes coordinates} if any reduced product of structures from said class witnesses a certain kind of rigidity phenomenon. We provide several equivalent characterizations of this property. This property has (at least) two remarkable consequences, one set-theoretic and one model-theoretic, for reduced products of structures of the said class. First, under appropriate set-theoretic assumptions every isomorphism between such reduced products associated with the Fréchet ideal lifts (modulo a finite change) to an isomorphism between products of the original structures. Second, with an additional mild assumption, it implies a strong quantifier elimination result. Of note, we show that a class recognizes coordinates if and only if an individual formula witnesses a certain syntactic property. We also consider many concrete classes of structures and determine whether or not they recognize coordinates. We place heavy emphasis on well-known classes of groups, such as permutation groups, acylindircally hyperbolic groups, quasisimple groups, free products, and graph products, but we also discuss other classes of structures.

math.LO↗