Related papers: Automorphism-invariant positive definite functions…
In this paper we look at the automorphisms of the multiplicative group of finite nearfields. We find partial results for the actual automorphism groups. We find counting techniques for the size of all finite nearfields. We then show that…
This is an expository article on properties of actions on Lie groups by subgroups of their automorphism groups. After recalling various results on the structure of the automorphism groups, we discuss actions with dense orbits, invariant and…
We give a survey on results regarding self-similar and automaton presentations of free groups and semigroups and related products. Furthermore, we discuss open problems and results with respect to algebraic decision problems in this area.
This is the first in a sequence of papers that will develop the theory of automorphisms of nonsolvable finite groups. The sequence will culminate in a new proof of McBride's Nonsolvable Signalizer Functor Theorem, which is one of the…
In this article we introduce powerful tools and techniques from invariant theory to free analysis. This enables us to study free maps with involution. These maps are free noncommutative analogs of real analytic functions of several…
A word in a free group is called ``potentially positive'' if it is automorphic to an element which is written with only positive exponents. We will develop automata to analyze properties of potentially positive words. We will use these to…
We investigate the concept of definable, or inner, automorphism in the logical setting of partial Horn theories. The central technical result extends a syntactical characterization of the group of such automorphisms (called the covariant…
In this article we present an extensive survey on the developments in the theory of non-abelian finite groups with abelian automorphism groups, and pose some problems and further research directions.
We discuss the set of subgroups of the automorphism group of a full shift, and submonoids of its endomorphism monoid. We prove closure under direct products in the monoid case, and free products in the group case. We also show that the…
We prove two conjectures on the automorphism group of a one-dimensional formal group law defined over a field of positive characteristic. The first is that if a series commutes with a nontorsion automorphism of the formal group law, then…
Using the theory of group action, we first introduce the concept of the automorphism group of an exponential family or a graphical model, thus formalizing the general notion of symmetry of a probabilistic model. This automorphism group…
In this paper, we discuss a group-theoretical generalization of the well-known Gauss formula involving the functionthat counts the number of automorphisms of a finite group. This gives several characterizations of finite cyclic groups.
We investigate the class of groups admitting an action on a set with an invariant mean. It turns out that many free products admit such an action. We give a complete characterisation of such free products in terms of a strong fixed point…
We present an effective algorithm for detecting automorphic orbits in free groups, as well as a number of algorithmic improvements of train tracks for free group automorphisms.
This paper describes the structure of invariant skew fields for linear actions of finite solvable groups on free skew fields in $d$ generators. These invariant skew fields are always finitely generated, which contrasts with the free algebra…
Let $G$ be a finitely generated group with an automorphism $\varphi\in{\rm Aut}(G)$, or an outer automorphism $\phi\in{\rm Out}(G)$. Suppose that $G$ decomposes into simpler pieces on which the growth behaviour of $\varphi$ and $\phi$ is…
We discuss the possibility of lifting finite subgroups, and in particular finite cyclic subgroups, with respect to the canonical projections between automorphism and outer automorphism groups of free groups, surface groups and their…
We will review the main results concerning the automorphism groups of saturated structures which were obtained during the two last decades. The main themes are: the small index property in the countable and uncountable cases; the…
We compute explicitly the automorphism and outer automorphism group of all large-type free-of-infinity Artin groups. Our strategy involves reconstructing the associated Deligne complexes in a purely algebraic manner, i.e. in a way that is…
This is a largely expository paper about how groups arise or are of interest in model theory. Included are the following topics: classifying groups definable in specific structures or theories and the relation to algebraic groups, groups…