Quotient Group Research Articles

Local finiteness and automorphism groups of low complexity subshifts

AbstractWe prove that for any transitive subshift X with word complexity function $c_n(X)$ , if $\liminf ({\log (c_n(X)/n)}/({\log \log \log n})) = 0$ , then the quotient group ${{\mathrm {Aut}(X,\sigma )}/{\langle \sigma \rangle }}$ of the automorphism group of X by the subgroup generated by the shift $\sigma $ is locally finite. We prove that significantly weaker upper bounds on $c_n(X)$ imply the same conclusion if the gap conjecture from geometric group theory is true. Our proofs rely on a general upper bound for the number of automorphisms of X of range n in terms of word complexity, which may be of independent interest. As an application, we are also able to prove that for any subshift X, if ${c_n(X)}/{n^2 (\log n)^{-1}} \rightarrow 0$ , then $\mathrm {Aut}(X,\sigma )$ is amenable, improving a result of Cyr and Kra. In the opposite direction, we show that for any countable infinite locally finite group G and any unbounded increasing $f: \mathbb {N} \rightarrow \mathbb {N}$ , there exists a minimal subshift X with ${{\mathrm {Aut}(X,\sigma )}/{\langle \sigma \rangle }}$ isomorphic to G and ${c_n(X)}/{nf(n)} \rightarrow 0$ .

Hyperbolic quotients of projection complexes

This paper is a continuation of our previous work with Margalit where we studied group actions on projection complexes. In that paper, we demonstrated sufficient conditions so that the normal closure of a family of subgroups of vertex stabilizers is a free product of certain conjugates of these subgroups. In this paper, we study both the quotient of the projection complex by this normal subgroup and the action of the quotient group on the quotient of the projection complex. We show that under certain conditions the quotient complex is $\\delta$-hyperbolic. Additionally, under certain circumstances, we show that if the original action on the projection complex was a non-elementary WPD action, then so is the action of the quotient group on the quotient of the projection complex. This implies that the quotient group is acylindrically hyperbolic.

On somep‐adic Galois representations and form class groups

Let $K$ be an imaginary quadratic field other than $\mathbb{Q}(\sqrt{-1})$ and $\mathbb{Q}(\sqrt{-3})$, and let $H_K$ be its Hilbert class field. For each prime $p$ we examine two $p$-adic abelian objects in view of class field theory. First, we investigate the image of a $p$-adic Galois representation of $\mathrm{Gal}(\overline{\mathbb{Q}}/H_K)$ attached to an elliptic curve with complex multiplication. Second, we construct a $p$-adic form class group that is isomorphic to a quotient group of $\mathrm{Gal}(K_{(p^\infty)}/H_K)$, where $K_{(p^\infty)}$ is the maximal abelian extension of $K$ unramified outside prime ideals lying above $p$.

Galois module structure of the units modulo $$p^m$$ of cyclic extensions of degree $$p^n$$

Let p be prime, and $$n,m \in \mathbb {N}$$ . When K/F is a cyclic extension of degree $$p^n$$ , we determine the $$\mathbb {Z}/p^m\mathbb {Z}[\text {Gal}(K/F)]$$ -module structure of $$K^\times /K^{\times p^m}$$ . With at most one exception, each indecomposable summand is cyclic and free over some quotient group of $$\text {Gal}(K/F)$$ . For fixed values of m and n, there are only finitely many possible isomorphism classes for the non-free indecomposable summand. These Galois modules act as parameterizing spaces for solutions to certain inverse Galois problems, and therefore this module computation provides insight into the structure of absolute Galois groups. More immediately, however, these results show that Galois cohomology is a context in which seemingly difficult module decompositions can practically be achieved: when $$m,n>1$$ the modular representation theory allows for an infinite number of indecomposable summands (with no known classification of indecomposable types), and yet the main result of this paper provides a complete decomposition over an infinite family of modules.

Cohomology rings of finite-dimensional pointed Hopf algebras over abelian groups

We show that the cohomology ring of a finite-dimensional complex pointed Hopf algebra with an abelian group of group-like elements is finitely generated. Our strategy has three major steps. We first reduce the problem to the finite generation of cohomology of finite dimensional Nichols algebras of diagonal type. For the Nichols algebras, we do a detailed analysis of cohomology via the Anick resolution reducing the problem further to specific combinatorial properties. Finally, to check these properties, we turn to the classification of Nichols algebras of diagonal type due to Heckenberger. In this paper, we complete the verification of these combinatorial properties for major parametric families, including Nichols algebras of Cartan and super types and develop all the theoretical foundations necessary for the case-by-case analysis. The remaining discrete families are addressed in a separate publication. As an application of the main theorem, we deduce finite generation of cohomology for other classes of finite-dimensional Hopf algebras, including basic Hopf algebras with abelian groups of characters and finite quotients of quantum groups at roots of one.

A simple solution of Stratton and Webb’s problem

The paper contains a simple solution of Stratton and Webb’s problem related to the so-called absolute annihilators of abelian groups researched also by Feigelstock. Namely, there are provided constructions of indecomposable torsion-free abelian groups G of any finite rank greater than two for which the quotient group modulo the subgroup generated by squares of all possible rings on G supports a non-zero associative ring structure. They are much less complicated than those obtained before and show that groups satisfying the mentioned condition are not as rare phenomena as might be supposed. Moreover, most of these constructions led to new examples of CR- and AR-groups, i.e. abelian groups supporting only commutative and only associative rings, respectively. In particular, the first examples of indecomposable as well as decomposable AR-groups which do not satisfy the condition CR are presented. The first example of an indecomposable torsion-free group satisfying the condition CR only in the class of associative rings is also obtained.

The Novikov conjecture and extensions of coarsely embeddable groups

Let 1\to N\to G\to G/N\to 1 be a short exact sequence of countable discrete groups and let B be any G - C^* -algebra. In this paper, we show that the strong Novikov conjecture with coefficients in B holds for such a group G when the normal subgroup N and the quotient group G/N are coarsely embeddable into Hilbert spaces. As a result, the group G satisfies the Novikov conjecture under the same hypothesis on N and G/N .

Paving tropical ideals

Tropical ideals are a class of ideals in the tropical polynomial semiring that combinatorially abstracts the possible collections of supports of all polynomials in an ideal over a field. We study zero-dimensional tropical ideals I with Boolean coefficients in which all underlying matroids are paving matroids, or equivalently, in which all polynomials of minimal support have support of size \(\deg (I)\) or \(\deg (I)+1\)—we call them paving tropical ideals. We show that paving tropical ideals of degree \(d+1\) are in bijection with \({\mathbb {Z}}^n\)-invariant d-partitions of \({\mathbb {Z}}^n\). This implies that zero-dimensional tropical ideals of degree 3 with Boolean coefficients are in bijection with \({\mathbb {Z}}^n\)-invariant 2-partitions of quotient groups of the form \({\mathbb {Z}}^n/L\). We provide several applications of these techniques, including a construction of uncountably many zero-dimensional degree-3 tropical ideals in one variable with Boolean coefficients, and new examples of non-realizable zero-dimensional tropical ideals.

When is the search of relatively maximal subgroups reduced to quotient groups?

Let $\mathfrak{X}$ be a class finite groups closed under taking subgroups, homomorphic images, and extensions, and let $\mathrm{k}_{\mathfrak{X}}(G)$ be the number of conjugacy classes $\mathfrak{X}$-maximal subgroups of a finite group $G$. The natural problem calling for a description, up to conjugacy, of the $\mathfrak{X}$-maximal subgroups of a given finite group is not inductive. In particular, generally speaking, the image of an $\mathfrak{X}$-maximal subgroup is not $\mathfrak{X}$-maximal in the image of a homomorphism. Nevertheless, there exist group homomorphisms that preserve the number of conjugacy classes of maximal $\mathfrak{X}$-subgroups (for example, the homomorphisms whose kernels are $\mathfrak{X}$-groups). Under such homomorphisms, the image of an $\mathfrak{X}$-maximal subgroup is always $\mathfrak{X}$-maximal, and, moreover, there is a natural bijection between the conjugacy classes of $\mathfrak{X}$-maximal subgroups of the image and preimage. In the present paper, all such homomorphisms are completely described. More precisely, it is shown that, for a homomorphism $\phi$ from a group $G$, the equality $\mathrm{k}_{\mathfrak{X}}(G)=\mathrm{k}_{\mathfrak{X}}(\operatorname{im} \phi)$ holds if and only if $\mathrm{k}_{\mathfrak{X}}(\ker \phi)=1$, which in turn is equivalent to the fact that the composition factors of the kernel of $\phi$ lie in an explicitly given list.

Properties of multi anti L-fuzzy quotient group A of a group G

Properties of multi anti L-fuzzy quotient group A of a group G

When is the search of relatively maximal subgroups reduced to quotient groups?

Пусть $\mathfrak{X}$ - класс конечных групп, замкнутый относительно подгрупп, гомоморфных образов и расширений, и $\mathrm{k}_{\mathfrak{X}}(G)$ - число классов сопряженности $\mathfrak{X}$-максимальных подгрупп конечной группы $G$. Естественная задача - описать с точностью до сопряженности $\mathfrak{X}$-максимальные подгруппы данной конечной группы - не индуктивна. В частности, в образе гомоморфизма образ $\mathfrak{X}$-максимальной подгруппы, вообще говоря, не $\mathfrak{X}$-максимален. Существуют гомоморфизмы, сохраняющие число классов сопряженности максимальных $\mathfrak{X}$-подгрупп (например, гомоморфизмы, ядра которых - $\mathfrak{X}$-группы). Относительно таких гомоморфизмов образ $\mathfrak{X}$-максимальной подгруппы всегда $\mathfrak{X}$-максимален и существует естественная биекция между классами сопряженности $\mathfrak{X}$-максимальных подгрупп образа и прообраза. В работе такие гомоморфизмы полностью описаны. Доказано, что для гомоморфизма $\phi$ из группы $G$ равенство $\mathrm{k}_{\mathfrak{X}}(G)=\mathrm{k}_{\mathfrak{X}}(\operatorname{im} \phi)$ выполнено, если и только если $\mathrm{k}_{\mathfrak{X}}(\ker \phi)=1$, а это, в свою очередь, равносильно тому, что композиционные факторы ядра $\phi$ принадлежат известному списку. Библиография: 25 наименований.

On the Extensions of Zassenhaus Lemma and Goursat’s Lemma to Algebraic Structures

The Jordan–Hölder theorem is proved by using Zassenhaus lemma which is a generalization of the Second Isomorphism Theorem for groups. Goursat’s lemma is a generalization of Zassenhaus lemma, it is an algebraic theorem for characterizing subgroups of the direct product of two groups G1 × G2, and it involves isomorphisms between quotient groups of subgroups of G1 and G2. In this paper, we first extend Goursat’s lemma to R‐algebras, i.e., give the version of Goursat’s lemma for algebras, and then generalize Zassenhaus lemma to rings, R‐modules, and R‐algebras by using the corresponding Goursat’s lemma, i.e., give the versions of Zassenhaus lemma for rings, R‐modules, and R‐algebras, respectively.

Relation between the left and right cosets of an α-normal subgroup

Let G be a group and α be an automorphism of G. In 2016, Ganjali and Erfanian introduced the notion of a normal subgroup related to α, called the α-normal subgroup. It is basically known that if N is an ordinary normal subgroup of G then every right coset Ng is actually the left coset gN. This fact allows us to define the product of two right cosets naturally, thus inducing the quotient group. This research investigates the relation between the left and right cosets of the relative normal subgroup. As we have done in the classic version, we then define the product of two right cosets in a natural way and continue with the construction of a, say, relative quotient group.

Investigation of Some Subgroups in Determining the Frattini Subgroup of Non-Abelian Finite Groups

Frattini subgroup, Φ(G), of a group G is the intersection of all the maximal subgroups of G, or else G itself if G has no maximal subgroups. If G is a p-group, then Φ(G) is the smallest normal subgroup N such the quotient group G/N is an elementary abelian group. It is against this background that the concept of p-subgroup and fitting subgroup play a significant role in determining Frattini subgroup (especially its order) of dihedral groups. A lot of scholars have written on Frattini subgroup, but no substantial relationship has so far been identified between the parent group G and its Frattini subgroup Φ(G) which this tries to establish using the approach of Jelten B. Napthali who determined some internal properties of non abelian groups where the centre Z(G) takes its maximum size.

A characterization of weakly Krull monoid algebras

Let D be a domain and let S be a torsion-free monoid such that D has characteristic 0 or the quotient group of S satisfies the ascending chain condition on cyclic subgroups. We give a characterization of when the monoid algebra D[S] is weakly Krull. As corollaries, we reobtain the results on when D[S] is Krull resp. weakly factorial, due to Chouinard resp. Chang. Furthermore, we deduce a characterization of generalized Krull monoid algebras analogous to our main result and we characterize the weakly Krull domains among the affine monoid algebras.

Some Neutrosophic Triplet Subgroup Properties and Homomorphism Theorems in Singular Weak Commutative Neutrosophic Extended Triplet Group

In 2018, the study of neutrosophic triplet cosets and neutrosophic triplet quotient group of a neutrosophic extended triplet group was initiated with a follow up of the establishment of fundamental homomorphism theorems for neutrosophic extended triplet group. But some lapses in these earlier results were identified and revised through the introduction of special kind of weak commutative neutrosophic extended triplet group (WCNETG) called perfect neutrosophic extended triplet group. Furthermore, neutro-homomorphism basic theorem has been established for commutative neutrosophic extended triplet group. In this current work, the generalization and extention of the above results was done by investigating neutro-homomorphism in singular WCNETG. This was achieved with the introduction and study of some new types of NT-subgroups that are right (left) cancellative, semi-strong, and maximally normal in a singular WCNETG. For any given non-empty subset S and NT-subgroup H of a singular WCNETG X, some of these new NT-subgroups were shown to exist as non-empty neutrosophic triplet normalizer, generated subset and centralizer of S, closure of H, derived subset of X and center of X. With these, the first, second and third neutro-isomorphism and neutro-correspondence theorems were established. This finally led to the proof of the neutro-Zassenhaus Lemma (Neutro-Butterfly Theorem).

The Group of Quotients of the Semigroup of Invertible Nonnegative Matrices Over Local Rings

In this paper, we prove that for a linearly ordered local ring R with 1/2 the group of quotients of the semigroup of invertible nonnegative matrices Gn(R) for n ≥ 3 coincides with the group GLn(R).

On the Cardinality of Relator Sets of Groups F/ ∏ [Ni, Nj

The present paper generalizes the results of our paper “On the finite presentability of the group F/[M, N]” to an arbitrary family of normal subgroups {Ni | i ∈ I} in a free group F. We obtain conditions for finite presentability of the quotient group F/ ∏ [Ni, Nj]. In both papers, the proof of the main result is based on the properties of verbal wreath products introduced by Alfred Lvovich Shmelkin.

The $(p,q,r)$-generations of the symplectic group $Sp(6,2)$

A finite group $G$ is called textit{$(l,m, n)$-generated}, if it is a quotient group of the triangle group $T(l,m, n) = left .$ In 29, Moori posed the question of finding all the $(p,q,r)$ triples, where $p, q$ and $r$ are prime numbers, such that a non-abelian finite simple group $G$ is a $(p,q,r)$-generated. In this paper we establish all the $(p,q,r)$-generations of the symplectic group $Sp(6,2).$ GAP 20 and the Atlas of finite group representations 33 are used in our computations.

A Novel Algebraic Structure of (α,β)-Complex Fuzzy Subgroups.

A complex fuzzy set is a vigorous framework to characterize novel machine learning algorithms. This set is more suitable and flexible compared to fuzzy sets, intuitionistic fuzzy sets, and bipolar fuzzy sets. On the aspects of complex fuzzy sets, we initiate the abstraction of -complex fuzzy sets and then define -complex fuzzy subgroups. Furthermore, we prove that every complex fuzzy subgroup is an -complex fuzzy subgroup and define -complex fuzzy normal subgroups of given group. We extend this ideology to define -complex fuzzy cosets and analyze some of their algebraic characteristics. Furthermore, we prove that -complex fuzzy normal subgroup is constant in the conjugate classes of group. We present an alternative conceptualization of -complex fuzzy normal subgroup in the sense of the commutator of groups. We establish the -complex fuzzy subgroup of the classical quotient group and show that the set of all -complex fuzzy cosets of this specific complex fuzzy normal subgroup form a group. Additionally, we expound the index of -complex fuzzy subgroups and investigate the -complex fuzzification of Lagrange’s theorem analog to Lagrange’ theorem of classical group theory.