Related papers: A Note on Generic Types

We prove an arithmetic regularity lemma for stable subsets of finite abelian groups, generalising our previous result for high-dimensional vector spaces over finite fields of prime order. A qualitative version of this generalisation was…

We prove that every finitely generated soluble group which is not virtually abelian has a subgroup of one of a small number of types.

We give a presentation of abelian class field theory.

Groups definable in simple theories retain the chain conditions and decomposition properties known from stable groups, up to commensurability. In the small case, if a generic type of G is not foreign to some type q, there is a q-internal…

We consider abelain subgroups of small index in finite groups. More generally, we consider subgroups such that the product of their index by the index of their centralizer is small.

We extend to semi-abelian categories the notion of characteristic subobject, which is widely used in group theory and in the theory of Lie algebras. Moreover, we show that many of the classical properties of characteristic subgroups of a…

This paper introduces the concept of slender generalized groups, extending the classical notion of slender abelian groups to the setting of generalized groups (completely simple semigroups). We establish fundamental properties of slender…

Given a finite abelian group $G$ and cyclic subgroups $A$, $B$, $C$ of $G$ of the same order, we find necessary and sufficient conditions for $A$, $B$, $C$ to admit a common transversal for the cosets they afford. For an arbitrary number of…

We classify smooth projective surfaces that are quotients of abelian surfaces by finite groups.

ECM survey article discussing the structure of subsets of Abelian groups which behave `a bit like' cosets (of subgroups).

We classify closed abelian subgroups of a compact simple Lie group of adjoint type and of type E having centralizer of the same dimension as the dimension of the subgroup and describe Weyl groups of maximal abelian subgroups.

We define the abelian fundamental group with modulus of a regular flat scheme over a discrete valuation ring, taking into account wild ramification along a divisor. Our definition provides a mixed-characteristic analogue of the abelian…

In this note some properties of the sum of element orders of a finite abelian group are studied.

We develop several aspects of local and global stability in continuous first order logic. In particular, we study type-definable groups and genericity.

In this note we show that groups with definable generics in a separably closed valued of finite imperfection degree can be embedded into groups definable in their algebraic closure.

We define here an analogue, for a semi-stable group scheme whose generic fiber is an abelian variety, of M. J. Taylor's class-invariant homomorphism (defined for abelian schemes), and we give a geometric description of it. Then we extend a…

The main goal of this paper is to apply the arithmetic method developed in our previous paper \cite{13} to determine the number of some types of subgroups of finite abelian groups.

In this paper we formalize some foundation concepts and theorems of group theory in a variant of type theory called the Calculus of Constructions with Definitions. In this theory we introduce definition of a group, which is both general and…

Adapting a proof of Bouscaren and Delon, we show that every type-definable connected group in a given stable theory of fields embeds into an algebraic group, under a condition on the definable closure. We also present general hypotheses…

A subgroup $H$ of a topological abelian group $X$ is said to be characterized by a sequence $\mathbf v =(v_n)$ of characters of $X$ if $H=\{x\in X:v_n(x)\to 0\ \text{in}\ \mathbb T\}$. We study the basic properties of characterized…