A permutation group G on a set A is ${\kappa }$ -homogeneous iff for all $X,Y\in \bigl [ {A} \bigr ]^ {\kappa } $ with $|A\setminus X|=|A\setminus Y|=|A|$ there is a $g\in G$ with $g[X]=Y$ . G is ${\kappa }$ -transitive iff for any injective function f with $\operatorname {dom}(f)\cup \operatorname {ran}(f)\in \bigl [ {A} \bigr ]^ {\le {\kappa }} $ and $|A\setminus \operatorname {dom}(f)|=|A\setminus \operatorname {ran}(f)|=|A|$ there is a $g\in G$ with $f\subset g$ .

Giving a partial answer to a question of P. M. Neumann [6] we show that there is an ${\omega }$ -homogeneous but not ${\omega }$ -transitive permutation group on a cardinal ${\lambda }$ provided

1. (i) ${\lambda }<{\omega }_{\omega }$ , or

2. (ii) $2^{\omega }<{\lambda }$ , and ${\mu }^{\omega }={\mu }^+$ and $\Box _{\mu }$ hold for each ${\mu }\le {\lambda }$ with ${\omega }=\operatorname {cf}({\mu })<{{\mu }}$ , or

3. (iii) our model was obtained by adding $(2^{\omega })^+$ many Cohen generic reals to some ground model.

For ${\kappa }>{\omega }$ we give a method to construct large ${\kappa }$ -homogeneous, but not ${\kappa }$ -transitive permutation groups. Using this method we show that there exist ${\kappa }^+$ -homogeneous, but not ${\kappa }^+$ -transitive permutation groups on ${\kappa }^{+n}$ for each infinite cardinal ${\kappa }$ and natural number $n\ge 1$ provided $V=L$ .

Let G be a permutation group on a set $\Omega $ of size t. We say that $\Lambda \subseteq \Omega $ is an independent set if its pointwise stabilizer is not equal to the pointwise stabilizer of any proper subset of $\Lambda $ . We define the height of G to be the maximum size of an independent set, and we denote this quantity $\textrm{H}(G)$ . In this paper, we study $\textrm{H}(G)$ for the case when G is primitive. Our main result asserts that either $\textrm{H}(G)< 9\log t$ or else G is in a particular well-studied family (the primitive large–base groups). An immediate corollary of this result is a characterization of primitive permutation groups with large relational complexity, the latter quantity being a statistic introduced by Cherlin in his study of the model theory of permutation groups. We also study $\textrm{I}(G)$ , the maximum length of an irredundant base of G, in which case we prove that if G is primitive, then either $\textrm{I}(G)<7\log t$ or else, again, G is in a particular family (which includes the primitive large–base groups as well as some others).

Is there a nontrivial automorphism of the Turing degrees? It is a major open problem of computability theory. Past results have limited how nontrivial automorphisms could possibly be. Here we consider instead how an automorphism might be induced by a function on reals, or even by a function on integers. We show that a permutation of ω cannot induce any nontrivial automorphism of the Turing degrees of members of 2ω, and in fact any permutation that induces the trivial automorphism must be computable.

A main idea of the proof is to consider the members of 2ω to be probabilities, and use statistics: from random outcomes from a distribution we can compute that distribution, but not much more.

Let $G$ be a finite group acting vertex-transitively on a graph. We show that bounding the order of a vertex stabiliser is equivalent to bounding the second singular value of a particular bipartite graph. This yields an alternative formulation of the Weiss conjecture.

We analyze flat ${{S}_{3}}$-covers of schemes, attempting to create structures parallel to those found in the abelian and triple cover theories. We use an initial local analysis as a guide in finding a global description.

We give a qualitative description of the set 𝒪G(H) of overgroups in G of primitive subgroups H of finite alternating and symmetric groups G, and particularly of the maximal overgroups. We then show that certain weak restrictions on the lattice 𝒪G(H) impose strong restrictions on H and its overgroup lattice.

While the classification project for the simple groups of finite Morley rank is unlikely to produce a classification of the simple groups of finite Morley rank, the enterprise has already arrived at a considerably closer approximation to that ideal goal than could have been realistically anticipated, with a mix of results of several flavors, some classificatory and others more structural, which can be combined when the stars are suitably aligned to produce results at a level of generality which, in parallel areas of group theory, would normally require either some additional geometric structure, or an explicit classification. And Bruno Poizat is generally awesome, though sometimes he goes too far.

For a permutation group H on an infinite set X and a transformation f of X, let 〈f: H〉 = 〈{hfh-1:h є; H}〉 be a group closure of f. We find necessary and sufficient conditions for distinct normal subgroups of the symmetric group on X and a one-to-one transformation f of X to generate distinct group closures of f. Amongst these group closures we characterize those that are left simple, left cancellative, idempotent-free semigroups, whose congruence lattice forms a chain and whose congruences are preserved under automorphisms.

This is a contribution to the study of line-transitive groups of automorphisms of finite linear spaces. Groups which are almost simple are of particular importance. In this paper almost simple line-transitive groups whose socle is an alternating group are classified. It is proved that the only alternating groups to occur are those of degrees 7 and 8, and that only one linear space occurs, namely a well-known space with 15 points and 35 lines. Although much of the proof exploits special properties of alternating groups, some general theory of groups acting line-transitively on finite linear spaces is developed.