Elementary Equivalence of Endomorphism Rings of Abelian p-Groups

Idempotent elements of the endomorphism semiring of a finite chain

Idempotents yield much insight in the structure of finite semigroups and semirings. In this article, we obtain some results on (multiplicatively) idempotents of the endomorphism semiring of a finite chain. We prove that the set of all idempotents with certain fixed points is a semiring and find its order. We further show that this semiring is an ideal in a well known semiring. The construction of an equivalence relation such that any equivalence class contain just one idempotent is proposed. In our main result we prove that such equivalence class is a semiring and find his order. We prove that…

Elementary equivalence of the automorphism groups of reduced Abelian p-groups

Consider unbounded reduced Abelian p-groups (p > 2) A and A'. In this paper, we prove that if the automorphism groups Aut A and Aut A' are elementary equivalent then the groups A and A' are equivalent in the second order logic bounded by the final rank of the basic subgroups of A and A'.

Elementary equivalence vs commensurability for hyperbolic groups

We study to what extent torsion-free (Gromov)-hyperbolic groups are elementarily equivalent to their finite index subgroups. In particular, we prove that a hyperbolic limit group either is a free product of cyclic groups and surface groups, or admits infinitely many subgroups of finite index which are pairwise non elementarily equivalent.

Elementary equivalence of Chevalley groups over fields

It is proved that (elementary) Chevalley groups $G_π(Φ,K)$ and $G_{π'}(Φ',K')$ (or $E_π(Φ,K)$ and $E_{π'}(Φ',K'))$ over infinite fields $K$ and $K'$ of characteristics $\ne 2$, with weight lattices $Λ$ and $Λ'$, respectively, are elementarily equivalent if and only if the root systems $Φ$ and $Φ'$ are isomorphic, the fields $K$ and $K'$ are elementarily equivalent, the lattices $Λ$ and $Λ'$ coincide.

On the endomorphism rings of abelian groups and their Jacobson radical

We give a characterization of those abelian groups which are direct sums of cyclic groups and the Jacobson radical of their endomorphism rings are closed. A complete characterization of $p$-groups $A$ for which $(EndA,\mathcal T_L)$ is locally compact, where $\mathcal T_L$ is the Liebert topology on $EndA$, is given. We prove that if $A$ is a countable elementary $p$-group then $EndA$ has a non-admissible ring topology. To every functorial topology on $A$ a right bounded ring topology on $EndA$ is attached. By using this topology we construct on $EndA$ a non-metrizable and non-admissibe ring t…

Elementary equivalence of right-angled Coxeter groups and graph products of finite abelian groups

We show that graph products of finite abelian groups are elementarily equivalent if and only if they are $\exists\forall$-equivalent if and only if they are isomorphic. In particular, two right-angled Coxeter groups are elementarily equivalent if and only if they are isomorphic.

Elementary equivalence in Artin groups of finite type

Irreducible Artin groups of finite type can be parametrized via their associated Coxeter diagrams into six sporadic examples and four infinite families, each of which is further parametrized by the natural numbers. Within each of these four infinite families, we investigate the relationship between elementary equivalence and isomorphism. For three out of the four families, we show that two groups in the same family are equivalent if and only if they are isomorphic; a positive, but weaker, result is also attained for the fourth family. In particular, we show that two braid groups are elementari…

Direct products and elementary equivalence of polycyclic-by-finite groups

We give an algebraic characterization of elementary equivalence for polycyclic-by-finite groups. Using this characterization, we investigate the relations between their elementary equivalence and the elementary equivalence of the factors in their decompositions in direct products of indecomposable groups. In particular we prove that the elementary equivalence of two such groups G,H is equivalent to each of the following properties: 1)Gx...xG (k times G) and Hx...xG (k times H) are elementarily equivalent for a strictly positive integer k; 2)AxG and AxH are elementarily equivalent for two eleme…

Kac-Moody 群的初等等价

Elementary equivalence of Kac-Moody groups

本文研究 Kac-Moody 群的初等等价，证明其蕴含广义 Cartan 矩阵、基域及孪生根数据的对应一致，并推广到有限域代数闭包的无穷子域情形。

自由阿贝尔乘曲面群中自同态的不动子群分类

Classification of Fixed Subgroups of Endomorphisms in Free-abelian Times Surface Groups

本文研究自由阿贝尔乘曲面群等群类的自同态性质，刻画其自同态双射条件，证明相关群的Hopf性，完成自同态不动子群完整分类并给出有限不动子群的等价条件。

Morita equivalence classes of blocks with elementary abelian defect groups of order 16

We classify the Morita equivalence classes of blocks with elementary abelian defect groups of order $16$ with respect to a complete discrete valuation ring with algebraically closed residue field of characteristic two. As a consequence, blocks with this defect group are derived equivalent to their Brauer correspondent in the normalizer of a defect group and so satisfy Broué's Conjecture.

Elementary equivalence of infinite-dimensional classical groups

Let D be a division ring such that the number of conjugacy classes in the multiplicative group D^* is equal to the power of D^*. Suppose that H(V) is the group GL(V) or PGL(V), where V is an infinite-dimensional vector space over D. We prove, in particular, that, uniformly in dim(V) and D, the first-order theory of H(V) is mutually syntactically interpretable with the theory of the two-sorted structure <dim(V),D> (whose only relations are the division ring operations on D) in the second-order logic with quantification over arbitrary relations of power <= dim(V). A certain analogue of this resu…

Diophantine Geometry over Groups X: The Elementary Theory of Free Products of Groups

This paper is the 10th in a sequence on the structure of sets of solutions to systems of equations over groups, projections of such sets (Diophantine sets), and the structure of definable sets over few classes of groups. In the 10th paper we answer affirmatively a question of R. L. Vaught on the elementary equivalence of free products of pairs of elementarily equivalent groups, and obtain a generalization of Tarski's problem on the elementary equivalence of non-abelian free groups. Finally, we prove that free products of stable groups is stable, generalizing a previous result on the stability …

An elementary approach to sofic equivalence relations

We present an elementary description of sofic equivalence relations, as well as some permanence properties for soficity. We answer a question by Conley, Kechris and Tucker-Drob of determining soficity in terms of the full group for aperiodic relations.

Quasi-inverse endomorphisms

Greither and Pareigis have established a connection between Hopf Galois structures on a Galois extension $L/K$ with Galois group $G$, and the regular subgroups of the group of permutations on $G$, which are normalized by $G$. Byott has rephrased this connection in terms of certain equivalence classes of injective morphisms of $G$ into the holomorph of the groups $N$ with the same cardinality of $G$. Childs and Corradino have used this theory to construct such Hopf Galois structures, starting from fixed-point-free endomorphisms of $G$ that have abelian images. In this paper we show that a fixed…

A topological zero-one law and elementary equivalence of finitely generated groups

Let $\mathcal G$ denote the space of finitely generated marked groups. We give equivalent characterizations of closed subspaces $\mathcal S\subseteq \mathcal G$ satisfying the following zero-one law: for any sentence $σ$ in the infinitary logic $\mathcal L_{ω_1, ω}$, the set of all models of $σ$ in $\mathcal S$ is either meager or comeager. In particular, we show that the zero-one law holds for certain natural spaces associated to hyperbolic groups and their generalizations. As an application, we obtain that generic torsion-free lacunary hyperbolic groups are elementarily equivalent; the same …

Morita equivalence classes of principal blocks with elementary abelian defect groups of order 64

We classify the Morita equivalence classes of principal blocks with elementary abelian defect groups of order 64 with respect to a complete discrete valuation ring with an algebraically closed residue field of characteristic two.

Morita equivalence classes of blocks with elementary abelian defect groups of order 32

We describe a general technique to classify blocks of finite groups, and we apply it to determine Morita equivalence classes of blocks with elementary abelian defect groups of order 32 with respect to a complete discrete valuation ring with an algebraically closed residue field of characteristic two. As a consequence we verify that a conjecture of Harada holds on these blocks.

光滑流形的初等等价与微分同胚群

Elementary equivalence and diffeomorphism groups of smooth manifolds

该研究证明了光滑流形的微分同胚群在初等等价下必然同胚，且在二维及以上维度中，保持光滑体积形式的微分同胚群也具有类似性质。

On the dynamics of extensions of free-abelian times free groups endomorphisms to the completion

We obtain conditions of uniform continuity for endomorphisms of free-abelian times free groups for the product metric defined by taking the prefix metric in each component and establish an equivalence between uniform continuity for this metric and the preservation of a coarse-median, which was recently introduced by Fioravanti. Considering the extension of an endomorphism to the completion we count the number of orbits for the action of the subgroup of fixed points (resp. periodic) points on the set of infinite fixed (resp. periodic) points. Finally, we study the dynamics of infinite points: f…

Finite self-similar p-groups with abelian first level stabilizers

We determine all finite p-groups that admit a faithful, self-similar action on the p-ary rooted tree such that the first level stabilizer is abelian. A group is in this class if and only if it is a split extension of an elementary abelian p-group by a cyclic group of order p. The proof is based on use of virtual endomorphisms. In this context the result says that if G is a finite p-group with abelian subgroup H of index p, then there exists a virtual endomorphism of G with trivial core and domain H if and only if G is a split extension of H and H is an elementary abelian p-group.

On the Gauge Equivalence of Twisted Quantum Doubles of Elementary Abelian and Extra-Special 2-Groups

We establish braided tensor equivalences among module categories over the twisted quantum double of a finite group defined by an extension of a group H by an abelian group, with 3-cocycle inflated from a 3-cocycle on H. We also prove that the canonical ribbon structure of the module category of any twisted quantum double of a finite group is preserved by braided tensor equivalences. We give two main applications: first, if G is an extra-special 2-group of width at least 2, we show that the quantum double of G twisted by a 3-cocycle w is gauge equivalent to a twisted quantum double of an elemen…

Broué's Conjecture for 2-blocks with elementary abelian defect groups of order 32

The first author has recently classified the Morita equivalence classes of 2-blocks B of finite groups with elementary abelian defect group of order 32. In all but three cases he proved that the Morita equivalence class determines the inertial quotient of B. We finish the remaining cases by utilizing the theory of lower defect groups. As a corollary, we verify Broué's Abelian Defect Group Conjecture in this situation.

procountable群的自同构拓扑同构是万有解析的

Topological isomorphism of procountable groups is universal analytic

本文证明procountable群的拓扑同构是万有解析等价关系，并推广至非阿基米德Polish群，通过编码权重和树构造实现分类复杂性。

Overgroups of elementary block-diagonal subgroups in the classical symplectic group over an arbitrary commutative ring

In this paper we prove a sandwich classification theorem for subgroups of the classical symplectic group over an arbitrary commutative ring $R$ that contain the elementary block-diagonal (or subsystem) subgroup $\operatorname{Ep}(ν, R)$ corresponding to a unitary equivalence realation $ν$ such that all self-conjugate equivalence classes of $ν$ are of size at least 4 and all not-self-conjugate classes of $ν$ are of size at least 5. Namely, given a subgroup $H$ of $\operatorname{Sp}(2n, R)$ such that $\operatorname{Ep}(ν, R) \le H$ we show that there exists a unique exact major form net of ideal…

伴随Chevalley群的Sha-刚性

Sha-rigidity of adjoint Chevalley groups of types $A_1$, $A_2$, $B_2$, $G_2$ over commutative rings

本文证明了在特定环上，A1、A2、B2和G2类型的伴随Chevalley群及其初等子群的局部内自同构是内自同构，从而这些群具有Sha-刚性。

On the Andrews-Curtis equivalence

The Andrews-Curtis conjecture claims that every balanced presentation of the trivial group can be reduced to the standard one by a sequence of ``elementary transformations" which are Nielsen transformations augmented by arbitrary conjugations. It is a prevalent opinion that this conjecture is false; however, not many potential counterexamples are known. In this paper, we show that some of the previously proposed examples are actually not counterexamples. We hope that the tricks we used in constructing relevant chains of elementary transformations will be useful to those who attempt to establis…

Homotopy equivalences between p-subgroup categories

Let p be a prime number and G a finite group of order divisible by p. Quillen showed that the Brown poset of nonidentity p-subgroups of G is homotopy equivalent to its subposet of nonidentity elementary abelian subgroups. We show here that a similar statement holds for the fusion category of nonidentity p-subgroups of G. Other categories of p-subgroups of G are also considered.