Related papers: Some remarks on dp-minimal groups

We show that $\omega$-categorical rings with NIP are nilpotent-by-finite. We prove that an $\omega$-categorical group with NIP and fsg is nilpotent-by-finite. We also notice that an $\omega$-categorical group with at least one strongly…

We prove that every \omega-categorical, generically stable group is nilpotent-by-finite and that every \omega-categorical, generically stable ring is nilpotent-by-finite.

A dp-minimal group is virtually nilpotent.

We explore the interplay between omega-categoricity and pseudofiniteness for groups, conjecturing that omega-categorical pseudofinite groups are finite-by-abelian-by-finite. We show that the conjecture reduces to nilpotent p-groups of class…

We prove that a finite-dimensional omega-categorical group is finite-by-abelian-by-finite and that a finite-dimensional omega-categorical ring is virtually finite-by-null.

Let $G$ be a dp-minimal group; we prove some consequences of several different hypotheses on $G$. First, if $G$ is torsion-free, then it is abelian. Second, if $G$ admits a distal f-generic type, then it is virtually nilpotent; we prove…

An $\omega$-categorical group of finite burden is virtually finite-by-abelian; an $\omega$-categorical ring of finite burden is virtually finite-by-null; an $\omega$-categorical NTP2 ring is virtually nilpotent.

We first show that every group-theoretical category is graded by a certain double coset ring. As a consequence, we obtain a necessary and sufficient condition for a group-theoretical category to be nilpotent. We then give an explicit…

The aim of this paper is to develop the theory of groups definable in the $p$-adic field ${\mathbb Q}_p$, with ``definable $f$-generics" in the sense of an ambient saturated elementary extension of ${\mathbb Q}_p$. We call such groups…

We study dp-minimal infinite profinite groups that are equipped with a uniformly definable fundamental system of open subgroups. We show that these groups have an open subgroup $A$ such that either $A$ is a direct product of countably many…

We show some basic facts about dp-minimal ordered structures. The main results are : dp-minimal groups are abelian-by-finite-exponent, in a divisible ordered dp-minimal group, any infinite set has non-empty interior, and any theory of pure…

Recall that a definable group is `definably amenable' if it admits a translation-invariant Keisler measure. We prove a combinatorial characterization of definable amenability for groups definable in NIP theories. More specifically, given a…

Let $G$ be an arbitrary group. We show that if the Fitting subgroup of $G$ is nilpotent then it is definable. We show also that the class of groups whose Fitting subgroup is nilpotent of class at most $n$ is elementary. We give an example…

Let $X$ be a definable group definable over a small model $M_0$. Recall that a global type $p$ on $X$ is definable $f$-generic over $M_0$ if every left translate of $p$ is definable over $M_0$. We call $p$ strongly $f$-generic over $M_0$ if…

We prove that each \omega-categorical, generically stable group is solvable-by-finite.

We define a notion of weak omega-category internal to a model of Martin-L\"of type theory, and prove that each type bears a canonical weak omega-category structure obtained from the tower of iterated identity types over that type. We show…

Let $A$ and $G$ be finite groups such that $A$ acts coprimely on $G$ by automorphisms. For any fixed prime divisor $p$ of $|G|$, we provide a complete characterization of the structure of a group $G$ in which every maximal $A$-invariant…

We show that every definable group G in an o-minimal structure is definably finitely generated. That is, G contains a finite subset that is not included in any proper definable subgroup. This provides another proof, and a generalization to…

We show that every finite group $G$ of size at least $3$ has a nilpotent subgroup of class at most $2$ and size at least $|G|^{1/32\log\log|G|}$. This answers a question of Pyber, and is essentially best possible.

We establish a surprising correspondence between groups definable in o-minimal structures and linear algebraic groups, in the nilpotent case. It turns out that in the o-minimal context, like for finite groups, nilpotency is equivalent to…