Comparing the automorphism group of the measure algebra with some groups related to the infinite permutation group of the natural numbers

Permutations of the integers induce only the trivial automorphism of the Turing degrees

Let $π$ be an automorphism of the Turing degrees induces by a homeomorphism $φ$ of the Cantor space $2^ω$ such that $φ$ preserves all Bernoulli measures. It is proved that $π$ must be trivial. In particular, a permutation of $ω$ can only induce the trivial automorphism of the Turing degrees.

Outer Automorphism Groups of Ordered Permutation Groups

An infinite linearly ordered set (S,<=) is called doubly homogeneous if its automorphism group A(S) acts 2-transitively on it. We show that any group G arises as outer automorphism group G cong Out(A(S)) of the automorphism group A(S), for some doubly homogeneous chain (S,<=).

传递扩展的通用结构自同构群

Transitive Extensions of Automorphism Groups of Generic Structures

研究传递扩展的存在性，发现仅当颜色数为2的幂时，边缘着色的k超图有传递扩展；k超 tournaments 在Cherlin意义上仅当k为偶数时存在传递扩展。

Infinite Limits of Finite-Dimensional Permutation Structures, and their Automorphism Groups: Between Model Theory and Combinatorics

In the course of classifying the homogeneous permutations, Cameron introduced the viewpoint of permutations as structures in a language of two linear orders, and this structural viewpoint is taken up here. The majority of this thesis is concerned with Cameron's problem of classifying the homogeneous structures in a language of finitely many linear orders, which we call finite-dimensional permutation structures. Towards this problem, we present a construction that we conjecture produces all such structures. Some evidence for this conjecture is given, including the classification of the homogene…

Twists and Twistability

Metrically homogeneous graphs are connected graphs which, when endowed with the path metric, are homogeneous as metric spaces. In this paper we introduce the concept of twisted automorphisms, a notion of isomorphism up to a permutation of the language. We find all permutations of the language which are associated with twisted automorphisms of metrically homogeneous graphs. For each non-trivial permutation of this type we also characterize the class of metrically homogeneous graphs which allow a twisted isomorphism associated with that permutation. The permutations we find are, remarkably, prec…

Permutation groups on countable vector spaces over prime fields

We describe all closed permutation groups which act on the set of vectors of a countable vector space $V$ over a prime field of odd order and which contain all automorphisms of $V$. In particular, we prove that their number is finite. These groups correspond, up to first-order interdefinability, precisely to all structures with a first-order definition in $V$.