Related papers: Some remarks on dp-minimal groups

We show that, for a finitely generated residually finite group $\Gamma$, the word $[x_1, \ldots, x_k]$ is a probabilistic identity of $\Gamma$ if and only if $\Gamma$ is virtually nilpotent of class less than $k$. Related results,…

A group is small if it has countably many complete $n$-types over the empty set for each natural number n. More generally, a group $G$ is weakly small if it has countably many complete 1-types over every finite subset of G. We show here…

Let $\mathfrak{Nil}$ be the class of nilpotent groups and $G$ be a group. We call $G$ a meta-$\mathfrak{Nil}$-Hamiltonian group if any of its non-$\mathfrak{Nil}$ subgroups is normal. Also, we call $G$ a para-$\mathfrak{Nil}$-Hamiltonian…

Let $\Gamma_G$ denote a graph associated with a group $G$. A compelling question about finite groups asks whether or not a finite group $H$ must be nilpotent provided $\Gamma_H$ is isomorphic to $\Gamma_G$ for a finite nilpotent group $G$.…

We prove a characterization of monomial projective representations of finitely generated nilpotent groups. We also characterize polycyclic groups whose projective representations are finite dimensional.

Let G be a powerful finite p-group. In this note, we give a short elementary proof of the following facts for all $i\ge 0$: (i) $\exp \Omega_-i(G)\le p^i$ for odd p, and $\exp \Omega_-i(G)\le 2^{i+1}$ for p = 2; (ii) the index $|G:G^{p^i}|$…

In this paper we find a characterization for groups elementarily equivalent to a free nilpotent group $G$ of class 2 and arbitrary finite rank.

We give an example of an omega-categorical group without AZ-enumerations. We build AZ-enumerations of some omega-categorical central products of countably many copies of a finite 2-step nilpotent group.

A finitely generated subgroup {\Gamma} of a real Lie group G is said to be Diophantine if there is \beta > 0 such that non-trivial elements in the word ball B_\Gamma(n) centered at the identity never approach the identity of G closer than…

We study definably amenable groups in NIP theories, and answer a question of Newelski (and also of Chernikov-Simon), by giving an example in the o-minimal context where weak generic types do not coincide with almost periodic types,…

We show that a finitely generated soluble group is virtually nilpotent if and only if the diameter of its finite coset spaces admits a uniform polynomial lower bound in terms of their size. We obtain the same conclusion for certain finitely…

We reformulate several basic notions of notions in finite group theory in terms of iterations of the lifting property (orthogonality) with respect to particular morphisms. Our examples include the notions being nilpotent, solvable, perfect,…

We prove a version of the Wilson conjecture for $\omega$-categorical $3$-Engel Lie algebras over a field of characteristic $5$: every $\omega$-categorical Lie algebra over $\mathbb{F}_5$ which satisfies the identity $[x,y^3] = 0$ is…

We introduce the notions of definable amenability and extreme definable amenability for groups in continuous structures and conduct an extensive analysis of them, drawing parallels with the classical first-order case. We characterize both…

We present a structural description of finite nilpotent groups of class at most $2$ using a specified number of subdirect and central products of $2$-generated such groups. As a corollary, we show that all of these groups are isomorphic to…

For $G$ a finite group, let $d_2(G)$ denote the proportion of triples $(x, y, z) \in G^3$ such that $[x, y, z] = 1$. We determine the structure of finite groups $G$ such that $d_2(G)$ is bounded away from zero: if $d_2(G) \geq \epsilon >…

A first order expansion of $(\mathbb{R},+,<)$ is dp-minimal if and only if it is o-minimal. We prove analogous results for algebraic closures of finite fields, $p$-adic fields, ordered abelian groups with only finitely many convex subgroups…

We prove generic differentiability in $P$-minimal theories, strengthening an earlier result of Kuijpers and Leenknegt. Using this, we prove Onshuus and Pillay's $P$-minimal analogue of Pillay's conjectures on o-minimal groups. Specifically,…

We present a complete list of groups $G$ and fields $F$ for which: (i) the group of normalized units V(FG) of the group algebra FG is locally nilpotent; (ii) the group algebra FG has a finite number of nilpotent elements and V(FG) is an…

In this paper we give a complete algebraic description of groups elementarily equivalent to a given free nilpotent group of finite rank.