Let p be an odd prime. A method is described which given a structure M of finite similarity type produces a nilpotent group of class 2 and exponent p which is in the same stability class as M. Theorem. There are nilpotent groups of class 2 and exponent p in all stability classes. Theorem. The problem of characterizing a stability class is equivalent to characterizing the (nilpotent, class 2, exponent p) groups in that class.

We show that an infinite field is interpretable in a stable torsion-free nilpotent groupG of classk, k>1. Furthermore we prove thatG/Z k-1 (G) must be divisible. By generalising methods of Belegradek we classify some stable torsion-free nilpotent groups modulo isomorphism and elementary equivalence.

We prove that for a finitely generated infinite nilpotent group $G$ with structure $(G,\cdot,\dots)$, the connected component ${G^*}^0$ of a sufficiently saturated extension $G^*$ of $G$ exists and equals \[ \bigcap_{n\in\N} \{g^n\colon g\in G^*\}. \] We construct an expansion of ${\mathbb Z}$ by a predicate $({\mathbb Z},+,P)$ such that the type-connected component ${{\mathbb Z}^*}^{00}_{\emptyset}$ is strictly smaller than ${{\mathbb Z}^*}^0$. We generalize this to finitely generated virtually solvable groups. As a corollary of our construction we obtain an optimality (...) result for the van der Waerden theorem for finite partitions of groups. (shrink)

In this paper we study the algebraic structure ofω-stable bilinear maps, arbitrary rings, and nilpotent groups. We will also provide rather complete structure theorems for the above structures in the finite Morley rank case.

In this paper, we give a complete algebraic description of groups elementarily equivalent to the P. Hall completion of a given free nilpotent group of finite rank over an arbitrary binomial domain. In particular, we characterize all groups elementarily equivalent to a free nilpotent group of finite rank.

We show that two abelian-by-finite groups are elementarily equivalent if and only if they satisfy the same sentences with two alternations of quantifiers. We also prove that abelian-by-finite groups satisfy a quantifier elimination property. On the other hand, for each integer n, we give some examples of nilpotent groups which satisfy the same sentences with n alternations of quantifiers and do not satisfy the same sentences with n + 1 alternations of quantifiers.

Let L\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${L}$$\end{document} be the language of group theory with n additional new constant symbols c1,…,cn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${c_1,\ldots,c_n}$$\end{document}. In L\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${L}$$\end{document} we consider the class K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb{K}}}$$\end{document} of all finite groups G of exponent p>2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${p > 2}$$\end{document}, where G′⊆⟨c1G,…,cnG⟩⊆Z\documentclass[12pt]{minimal} \usepackage{amsmath} (...) \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${G'\subseteq\langle c_1^G,\ldots,c_n^G\rangle \subseteq Z}$$\end{document} and c1G,…,cnG\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${c_1^G,\ldots,c_n^G}$$\end{document} are linearly independent. Using amalgamation we show the existence of Fraïssé limits D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${D}$$\end{document} of K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb{K}}}$$\end{document}. D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${D}$$\end{document} is Felgner’s extra special p-group. The elementary theories of the D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${D}$$\end{document} are supersimple of SU-rank 1. They have the independence property. (shrink)

A well known and referenced global result is the nilpotent characterisation of the finite p-groups. This un doubtedly transends into neutrosophy. Hence, this fact of the neutrosophic nilpotent p-groups is worth critical studying and comprehensive analysis. The nilpotent characterisation depicts that there exists a derived series (Lower Central) which must terminate at {ϵ} (an identity), after a finite number of steps. Now, Suppose that G(I) is a neutrosophic p-group of class at least m ≥ 3. (...) We show in this paper that Lm−1(G(I)) is abelian and hence G(I) possesses a characteristic abelian neutrosophic subgroup which is not supposed to be contained in Z(G(I)). Furthermore, If L3(G(I)) = 1 such that pm is the highest order of an element of G(I)/L2(G(I)) (where G(I) is any neutrosophic p-group) then no element of L2(G(I)) has an order higher than pm. (shrink)

The following theorems are proved about the Frattini-free componentG Φ of a soluble stable ℜ-group: a) If it has a normal subgroupN with nilpotent quotientG Φ/N, then there is a nilpotent subgroupH ofG Φ withG Φ=NH. b) It has Carter subgroups; if the group is small, they are all conjugate. c) Nilpotency modulo a suitable Frattini-subgroup (to be defined) implies nilpotency. The last result makes use of a new structure theorem for the centre of the derivative of the (...) Frattini-free component of a centreless soluble ℜ-group. (shrink)

We prove that in a nilpotent-by-finite stable group an equation that holds generically holds everywhere. Combining this result with results of Wagner and Bryant, we conclude that a soluble-by-finite stable group of generic exponent n has exponent n.

RésuméOn prouve que dans un groupe stable, nilpotent par fini, une équation générìquement satisfaite y est partout satisfaite. En combinant ce résultat avec des résultats de Wagner et Bryant, on déduit qu'un groupe stable résoluble-par-fini d'exposant généríquenest d'exposantn.

The terms of the upper and lower central series of a nilpotent computable group have computably enumerable Turing degree. We show that the Turing degrees of these terms are independent even when restricted to groups which admit computable orders.

A group is small if it has only 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 only countably many complete 1-types over every finite subset of G. We show here that in a weakly small group, subgroups which are definable with parameters lying in a finitely generated algebraic closure satisfy the descending chain conditions for their traces in any finitely generated algebraic closure. An infinite (...) weakly small group has an infinite abelian subgroup, which may not be definable. A small nilpotent group is the central product of a definable divisible group with a definable one of bounded exponent. In a group with simple theory, any set of pairwise commuting elements is contained in a definable finite-by-abelian subgroup. First corollary: a weakly small group with simple theory has an infinite definable finite-by-abelian subgroup. Secondly, in a group with simple theory, a solvable group A of derived length n is contained in an A-definable almost solvable group of class at most 2n — 1. (shrink)

It is known that the spaces of orders on orderable computable fields can represent all Π10 classes up to Turing degree. We show that the spaces of orders on orderable computable abelian and nilpotent groups cannot represent Π10 classes in even a weak manner. Next, we consider presentations of ordered abelian groups, and we show that there is a computable ordered abelian group for which no computable presentation admits a computable set of representatives for its Archimedean classes.

We prove the following theorem. Let G be a connected simple bad group (i.e. of finite Morley rank, nonsolvable and with all the Borel subgroups nilpotent) of minimal Morley rank. Then the Borel subgroups of G are conjugate to each other, and if B is a Borel subgroup of G, then $G = \bigcup_{g \in G}B^g,N_G(B) = B$ , and G has no involutions.

We know quite a lot about the general structure of ω-stable solvable centerless groups of finite Morley rank. Abelian groups of finite Morley rank are also well-understood. By comparison, nonabelian nilpotent groups are a mystery except for the following general results:• An ω1-categorical torsion-free nonabelian nilpotent group is an algebraic group over an algebraically closed field of characteristic 0 [Z3].• A nilpotent group of finite Morley rank is the central product of a definable subgroup (...) of finite exponent and of a definable divisible subgroup [N3].• A divisible nilpotent group of finite Morley rank is the direct product of its torsion part (which is central) and of a torsion-free subgroup [N3].However, we do not understand nilpotent groups of bounded exponent. It seems that the classification of nilpotent (but nonabelian)p-groups of finite Morley rank is impossible. Even the nilpotent groups of Morley rank 2 contain insurmountable difficulties [C], [T]. At first glance, this may seem to be an obstacle to proving the Cherlin-Zil'ber conjecture (“simple groups of finite Morley rank are algebraic groups”). Our purpose in this article is to show that if such a group is a definable subgroup of a nonnilpotent group, then it is possible to obtain a classification within the boundaries of our present knowledge. In this respect, our article may be considered as a relief to those who are trying to classify simple groups of finite Morley rank.Before explicitly stating our result, we need the following definition. (shrink)

RésuméACarter subgroupis a self-normalizing locally nilpotent subgroup. For studying these subgroups in groups of finite Morley rank, we introduce the new notion of alocally closed subgroup. We show that every solvable group of finite Morley rank has a unique conjugacy class of Carter subgroups.

A Carter subgroup is a self-normalizing locally nilpotent subgroup. For studying these subgroups in groups of finite Morley rank, we introduce the new notion of a locally closed subgroup. We show that every solvable group of finite Morley rank has a unique conjugacy class of Carter subgroups.

We study theorems of ordered groups from the perspective of reverse mathematics. We show that suffices to prove Hölder's Theorem and give equivalences of both (the orderability of torsion free nilpotent groups and direct products, the classical semigroup conditions for orderability) and (the existence of induced partial orders in quotient groups, the existence of the center, and the existence of the strong divisible closure).

Given a pseudovariety [MATHEMATICAL SCRIPT CAPITAL C], it is proved that a residually-[MATHEMATICAL SCRIPT CAPITAL C] superstable group G has a finite seriesG0 ⊴ G1 ⊴ · · · ⊴ Gn = Gsuch that G0 is solvable and each factor Gi +1/Gi is in [MATHEMATICAL SCRIPT CAPITAL C] . In particular, a residually finite superstable group is solvable-by-finite, and if it is ω -stable, then it is nilpotent-by-finite. Given a finitely generated group G, we show that if G is (...) ω -stable and satisfies some residual properties , then G is finite. (shrink)

Si C est une pseudo-variété, alors un groupe supersimple résiduellement C est nilpotent-par-poly-C. If C is a pseudo-variety, then a supersimple residually C group is nilpotent-by-poly-C.

We develop a basic theory of rosy groups and we study groups of small Uþ-rank satisfying NIP and having finitely satisfiable generics: Uþ-rank 1 implies that the group is abelian-by-finite, Uþ-rank 2 implies that the group is solvable-by-finite, Uþ-rank 2, and not being nilpotent-by-finite implies the existence of an interpretable algebraically closed field.

We are concerned with identifying by how much a finite cover of an 0-categorical structure differs from a sequence of free covers. The main results show that this is measured by automorphism groups which are nilpotent-by-abelian. In the language of covers, these results say that every finite cover can be decomposed naturally into linked, superlinked and free covers. The superlinked covers arise from covers over a different base, and to describe this properly we introduce the notion of a (...) quasi-cover.These results generalise results of the second author obtained in the case where the base of the cover is a grassmannian of a disintegrated set. They also give a complete proof of a statement of the second author extending this case to the case of a grassmannian of a modular set. To do this, we need to analyse the possible superlinked covers of such a set.We also give a combinatorial condition on the base of a cover which guarantees various chain conditions on finite covers over this base, and introduce a pregeometry which is useful in the analysis of finite covers with simple fibre groups. (shrink)

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

We define a generalized version of CM-triviality, and show that in the presence of enough regular types, or solubility, a stable CM-trivial group is nilpotent-by-finite. A torsion-free small CM-trivial stable group is abelian and connected. The first result makes use of a generalized version of the analysis of bad groups.

The neutrosophic automorphisms of a neutrosophic groups G (I) , denoted by Aut(G (I)) is a neu-trosophic group under the usual mapping composition. It is a permutation of G (I) which is also a neutrosophic homomorphism. Moreover, suppose that X1 = X(G (I)) is the neutrosophic group of inner neutrosophic auto-morphisms of a neutrosophic group G (I) and Xn the neutrosophic group of inner neutrosophic automorphisms of Xn-1. In this paper, we show that if any neutrosophic group of the (...) sequence G (I), X1, X2, … is the identity, then G (I) is nilpotent. (shrink)

The paper continues [1]. Let S be a complete theory of ultraflat (e.g. planar) graphs as introduced in [4]. We show a strong form of NOTOP for S: The union of two models M1 and M2, independent over a common elementary submodel M0, is the primary model over M1 ∪ M2 of S. Then by results of [1] Mekler's construction [6] gives for such a theory S of nice ultraflat graphs a superstable 2-step-nilpotent group of exponent $p (>2)$ with (...) NDOP and NOTOP. (shrink)

If $G$ is a group with a supersimple theory having a finite $SU$-rank, then the subgroup of $G$ generated by all of its normal nilpotent subgroups is definable and nilpotent. This answers a question asked by Elwes, Jaligot, Macpherson and Ryten. If $H$ is any group with a supersimple theory, then the subgroup of $H$ generated by all of its normal soluble subgroups is definable and soluble.

The notion of CM-triviality was introduced by Hrushovski, who showed that his new strongly minimal sets have this property. Recently Baudisch has shown that his new ω 1 -categorical group has this property. Here we show that any group of finite Morley rank definable in a CM-trivial theory is nilpotent-by-finite, or equivalently no simple group of finite Morley rank can be definable in a CM-trivial theory.

Let G be a finite group. For every formula ø in the language of groups, let K denote the class of groups H such that ø is a normal abelian subgroup of H and the quotient group H;ø is isomorphic to G. We show that if G is nilpotent and its order is not square-free, then there exists a formula ø such that the theory of K is undecidable.

We investigate the structure of C-minimal valued groups that are not abelian-by-finite. We prove among other things that they are nilpotent-by-finite.

We show that a stable groupG satisfying certain commutator conditions is nilpotent. Furthermore, a soluble stable group with generically splitting automorphism of prime order is nilpotent-by-finite. In particular, a soluble stable group with a generic element of prime order is nilpotent-by-finite.

It is known that there exists a first-order sentence that holds in a finite group if and only if the group is soluble. Here it is shown that the corresponding statements with ‘solubility’ replaced by ‘nilpotence’ and ‘perfectness’, among others, are false.These facts present difficulties for the study of pseudofinite groups. However, a very weak form of Frattini’s theorem on the nilpotence of the Frattini subgroup of a finite group is proved for pseudofinite groups.

We consider whether given a simple, finite description of a group in the form of an algorithm, it is possible to algorithmically determine if the corresponding group has some specified property or not. When there is such an algorithm, we say the property is recursively recognizable within some class of descriptions. When there is not, we ask how difficult it is to detect the property in an algorithmic sense. We consider descriptions of two sorts: first, recursive presentations in terms of (...) generators and relators, and second, algorithms for computing the group operation. For both classes of descriptions, we show that a large class of natural algebraic properties, Markov properties, are not recursively recognizable, indeed they are \-hard to detect in recursively presented groups and \-hard to detect in computable groups. These theorems suffice to give a sharp complexity measure for the detection problem of a number of typical group properties, for example, being abelian, torsion-free, orderable. Some properties, like being cyclic, nilpotent, or solvable, are much harder to detect, and we give sharp characterizations of the corresponding detection problems from a number of them. We give special attention to orderability properties, as this was a main motivation at the beginning of this project. (shrink)

Biology is seen not merely as a privileged oppressor of women but as a co-victim of masculinist social assumptions. We see feminist critique as one of the normative controls that any scientist must perform whenever analyzing data, and we seek to demonstrate what has happened when this control has not been utilized. Narratives of fertilization and sex determination traditionally have been modeled on the cultural patterns of male/female interaction, leading to gender associations being placed on cells and their components. We (...) also find that when gender biases are controlled, new perceptions of these intracellular and extracellular relationships emerge. (shrink)

Since Darwin, the idea of psychological continuity between humans and other animals has dominated theory and research in investigating the minds of other species. Indeed, the field of comparative psychology was founded on two assumptions. First, it was assumed that introspection could provide humans with reliable knowledge about the causal connection between specific mental states and specific behaviors. Second, it was assumed that in those cases in which other species exhibited behaviors similar to our own, similar psychological causes were at (...) work. In this paper. we show how this argument by analogy is flowed with respect to the case of second-order mental states. As a test case, we focus on the question of how other species conceive of visual attention, and in particular whether chimpanzees interpret seeing as a mentalistic event involving internal states of perception, attention, and belief. We conclude that chimpanzees do not reason about seeing in this manner, and indeed, there is considerable reason to suppose that they do not harbor representations of mental states in general. We propose a reinterpretation model in which the majority of the rich social behaviors that humans and other primates share in common emerged long before the human lineage evolved the psychological means of interpreting those behaviors in mentalistic terms. Although humans, chimpanzees, and most other species may be said to possess mental states, humans alone may have evolved a cognitive specialization for reasoning about such states. (shrink)

En 1983, un groupe de femmes japonaises francophiles, très intéressées par le féminisme français et les changements rapides de la vie des femmes en France depuis les années 1970, créent la Société Franco-japonaise des Études sur les Femmes. Le but est de promouvoir une meilleure compréhension entre les deux cultures et de développer des liens avec les Françaises sur un grand nombre de questions féminines. Le projet se révéla vite être un succès. Parmi les études entreprises par la Société, l’analyse (...) comparée de l’histoire des Françaises et des Japonaises, domaine très peu exploré alors, a suscité un vif intérêt chez les participantes. Cet article revient sur vingt années de réflexion, de recherche, et d’échange par un groupe d’historiennes non-professionnelles sur les relations entre histoire et expériences de vie personnelles. Il nous montre comment l’étude de l’histoire des femmes dans un contexte international peut être fructueuse et bénéfique, et combien une libre discussion collective, sans tabous, sur ce sujet peut éveiller ou approfondir la conscience féministe. (shrink)

Malm and colleagues (2008) consider (and reject) five arguments putatively justifying the idea that healthcare workers (HCWs) have a duty to treat (DTT) during a pandemic. We do not have sufficient...

We would like to open some questions here about the institutional and cultural conditions of anything that might be called cultural studies or multiculturalism. By introducing cultural studies and multiculturalism many intellectuals aim at a more democratic culture. We share this aim. In this essay, however, we would like to argue that the projects of cultural studies and multiculturalism require: a more international model of cultural studies than the dominant Anglo-American versions; renewed attention to the institutional environments of cultural studies; (...) and a questioning of the relations between multiculturalism and identity politics. We seek less to “fix” these problems than to provide a critical analysis of the languages, the methods of criticism, and the assumptions about identity, culture, and politics that present the problems to us. Because the thickets entangling what our group calls cultural studies are so deeply rooted in Western academia, which to a large degree constitutes our own group, the counterexample of cultural criticism in other contexts can be more than usually instructive. We begin by considering the position of cultural studies in China, since our group includes a number of Chinese intellectuals, on whose experience the following section is largely based. The Chicago Cultural Studies Group began meeting in June 1990. It includes Lauren Berlant, David Bunn, Vinay Dharwadker, Norma Field, Dilip Gaonkar, Marilyn Ivy, Benjamin Lee, Leo Ou-fan Lee, Xinmin Liu, Mathew Roberts, Sharon Stephens, Katie Trumpener, Greg Urban, Michael Warner, Jianyang Zha, and Jueliang Zhou. (shrink)

En 1983, un groupe de femmes japonaises francophiles, très intéressées par le féminisme français et les changements rapides de la vie des femmes en France depuis les années 1970, créent la Société Franco-japonaise des Études sur les Femmes. Le but est de promouvoir une meilleure compréhension entre les deux cultures et de développer des liens avec les Françaises sur un grand nombre de questions féminines. Le projet se révéla vite être un succès. Parmi les études entreprises par la Société, l’analyse (...) comparée de l’histoire des Françaises et des Japonaises, domaine très peu exploré alors, a suscité un vif intérêt chez les participantes. Cet article revient sur vingt années de réflexion, de recherche, et d’échange par un groupe d’historiennes non-professionnelles sur les relations entre histoire et expériences de vie personnelles. Il nous montre comment l’étude de l’histoire des femmes dans un contexte international peut être fructueuse et bénéfique, et combien une libre discussion collective, sans tabous, sur ce sujet peut éveiller ou approfondir la conscience féministe. (shrink)