Related papers: Some remarks on dp-minimal groups

Given a definably compact group G in a saturated o-minimal structure, there is a canonical homomorphism from G to a compact real Lie group F(G). We establish a similar result for the (o-mininimal) universal cover of a definably compact…

We provide an explicit construction for a complete set of orthogonal primitive idempotents of finite group algebras over nilpotent groups. Furthermore, we give a complete set of matrix units in each simple epimorphic image of a finite group…

We prove that if $(H,G)$ is a small, $nm$-stable compact $G$-group, then $H$ is nilpotent-by-finite, and if additionally $\NM(H) \leq \omega$, then $H$ is abelian-by-finite. Both results are significant steps towards the proof of the…

We prove in particular that, in a large class of dp-minimal theories including the p-adics, definable types are dense amongst non-forking types.

This paper is about the $dfg$/$fsg$ decomposition for groups $G$ definable in $p$-adically closed fields. It is proved that for $G$ definably amenable, $G$ has a definable normal $dfg$ subgroup $H$ such that the quotient $G/H$ is a…

A finite $p$-group $G$ is said to be $d$-maximal if $d(H)<d(G)$ for every subgroup $H<G$, where $d(G)$ denotes the minimal number of generators of $G$. A similar definition can be formulated when $G$ is acted on by some group $A$. We…

We study infinite groups interpretable in three families of valued fields: $V$-minimal, power bounded $T$-convex, and $p$-adically closed fields. We show that every such group $G$ has unbounded exponent and that if $G$ is dp-minimal then it…

The class of all subdirectly irreducible groups belonging to a variety generated by a finite nilpotent group can be axiomatised by a finite set of elementary sentences.

Let $g$ be an element of a group $G$. For a positive integer $n$, let $E_n(g)$ be the subgroup generated by all commutators $[...[[x,g],g],\dots ,g]$ over $x\in G$, where $g$ is repeated $n$ times. We prove that if $G$ is a profinite group…

In this note we show that for any powerful $p$-group $G$, the subgroup $\Omega_{i}(G^{p^{j}})$ is powerfully nilpotent for all $i,j\geq1$ when $p$ is an odd prime, and $i\geq1$, $j\geq2$ when $p=2$. We provide an example to show why this…

We study the notion of dp-minimality, beginning by providing several essential facts, establishing several equivalent definitions, and comparing dp-minimality to other minimality notions. The rest of the paper is dedicated to examples. We…

In the first part, we prove that the dominion (in the sense of Isbell) of a subgroup of a finitely generated nilpotent group is trivial in the category of all nilpotent groups. In the second part, we show that the dominion of a subgroup of…

Let G be a unipotent algebraic group over an algebraically closed field k of characteristic p > 0 and let l be a prime different from p. Let e be a minimal idempotent in D_G(G), the braided monoidal category of G-equivariant (under…

We continue our local analysis of groups interpretable in various dp-minimal valued fields, as introduced in [8]. We associate with every infinite group $G$ interpretable in those fields an infinite type-definable infinitesimal subgroup…

We show that an infinite group $G$ definable in a $1$-h-minimal field admits a strictly $K$-differentiable structure with respect to which $G$ is a (weak) Lie group, and show that definable local subgroups sharing the same Lie algebra have…

It is shown that finite groups in which the order of the product of every pair of elements of co-prime order is the product of the orders, is nilpotent.

It is proved that the derived subgroup of a finite group is nilpotent if and only if $|ab|\ge |a||b|$ for all primary commutators $a$ and $b$ of coprime orders.

We answer the question, raised more than thirty years ago, on whether the power (G raised to the power omega) of a countably compact minimal Abelian group G is minimal, by showing that the negative answer is equivalent to the existence of…

We describe maximal nilpotent subsemigroups of a given nilpotency class in the semigroup $\Omega_n$ of all $n\times n$ real matrices with non-negative coefficients and the semigroup $\mathbf{D}_n$ of all doubly stochastic real matrices.

Let $p$ be a prime number and suppose that every maximal subgroup of a finite group is either $p$-nilpotent or has prime index. Such group need not be $p$-solvable, and we study its structure by proving that only one nonabelian simple group…