Left Cosets Research Articles

The Category G-GrR-Mod and Group Factorization

In this work, we use the concept of G-weak graded rings and G-weak graded modules, which are based on grading by a set G of left coset representatives for the left action of a subgroup H of a finite X on X, to define the conjugation action of the set G and to generalize and prove some results from the literature. In particular, we prove that a G-weak graded ring R is strongly graded if and only if each G-weak graded R-module V is induced by an ReG-module. Moreover, we prove that the additive induction functor (−)R and the restriction functor (−)eG form an equivalence between the categories G-GrR-Mod and ReG-Mod when R is strongly G-weak graded. Furthermore, some related results and illustrative examples of G-weak graded R-modules and their morphisms are provided.

Contracting boundary of a cusped space

Let [Formula: see text] be a finitely generated group. Cashen and Mackay proved that if the contracting boundary of [Formula: see text] with the topology of fellow traveling quasi-geodesics is compact then [Formula: see text] is a hyperbolic group. Let [Formula: see text] be a finite collection of finitely generated infinite index subgroups of [Formula: see text]. Let [Formula: see text] be the cusped space obtained by attaching combinatorial horoballs to each left coset of elements of [Formula: see text]. In this article, we prove that if the combinatorial horoballs are contracting and [Formula: see text] has compact contracting boundary then [Formula: see text] is hyperbolic relative to [Formula: see text].

Commuting Probability for Approximate Subgroups of a Finite Group

Abstract For subsets $X,Y$ of a finite group G, we write $\mathrm{Pr} (X,Y)$ for the probability that two random elements $x\in X$ and $y\in Y$ commute. This paper addresses the relation between the structure of an approximate subgroup $A\subseteq G$ and the probabilities $\mathrm{Pr} (A,G)$ and $\mathrm{Pr} (A,A)$. The following results are obtained. Theorem 1.1: Let A be a K-approximate subgroup of a finite group G, and let $\mathrm{Pr} (A,G)\geq\epsilon\gt0$. There are two $(\epsilon, K)$-bounded positive numbers γ and K0 such that G contains a normal subgroup T and a K0-approximate subgroup B such that $|A\cap B|\ge \gamma\; {\mathrm{max}}\{|A|,|B|\}$ while the index $[G:T]$ and the order of the commutator subgroup $[T,\langle B \rangle]$ are $(\epsilon, K)$-bounded. Theorem 1.2: Let A be a K-approximate subgroup of a finite group G, and let $\mathrm{Pr} (A,A)\geq\epsilon\gt0$. There are two $(\epsilon, K)$-bounded positive numbers γ and s and a subgroup $C\leq G$ such that $|C \cap A^2| \gt \gamma |A|$ and $|C^{\prime}|\leq s$. In particular, A is contained in the union of at most $\gamma^{-1}K^2$ left cosets of the subgroup C. It is also shown that the above results admit approximate converses.

Graded Rings Associated with Factorizable Finite Groups

Let R be an associative ring with unity, X be a finite group, H be a subgroup of X, and G be a set of left coset representatives for the left action of H on X. In this article, we introduce two different ways to put R into a non-trivial G-weak graded ring that is a ring graded by the set G which is defined with a binary operation ∗ and satisfying an algebraic structure with specific properties. The first one is by choosing a subset S of G such that S is a group under the ∗ operation and putting Rt=0 for all t∈G and t∉S. The second way, which is the most important, is induced by combining the operation ∗ defined on G and the coaction ◁ of H on G. Many examples are provided.

SIFAT SUBGRUP NORMAL DARI ANTI SUBGRUP NORMAL FUZZY

A fuzzy set is a concept theory that provide a solution of problem that cannot be explained by crisp set. Along with time, research of a fuzzy set are combined with algebra that produce fuzzy algebra. One of the research is a fuzzy subgroup and fuzzy level subset. The other research is an anti fuzzy subgroup that is inspired by a fuzzy subgroup. The purpose of this research is to write further study of anti fuzzy subgroup properties by induction of properties of fuzzy algebra such as fuzzy set, fuzzy subgroup, and anti fuzzy subgroup. The research procedure is to study the basic concept of fuzzy set, fuzzy subgroup, and anti fuzzy subgroup. Using that concept to proof the properties of the anti fuzzy subgroup. The conclusions are in the anti fuzzy subgroup, set with anti fuzzy subgroup is a subgroup of a group that can be applied to complement of and normal anti fuzzy subgroup closely related to anti fuzzy left coset, right coset, and middle coset.

Computations for the groups SL(2,81) and SL(2,729)

For any group G, we define G/H (read” G mod H”) to be the set of left cosets of H in G and this set forms a group under the operation (a)(bH) = abH. The character table of rational representations study to gain the K( SL(2,81)) and K( SL(2, 729)) in this work.

Quasi-isometric rigidity of subgroups and filtered ends

Let $G$ and $H$ be quasi-isometric finitely generated groups and let $P\leq G$; is there a subgroup $Q$ (or a collection of subgroups) of $H$ whose left cosets coarsely reflect the geometry of the left cosets of $P$ in $G$? We explore sufficient conditions for a positive answer. The article consider pairs of the form $(G,\mathcal{P})$ where $G$ is a finitely generated group and $\mathcal{P}$ a finite collection of subgroups, there is a notion of quasi-isometry of pairs, and quasi-isometrically characteristic collection of subgroups. A subgroup is qi-characteristic if it belongs to a qi-characteristic collection. Distinct classes of qi-characteristic collections of subgroups have been studied in the literature on quasi-isometric rigidity, we list in the article some of them and provide other examples. The first part of the article proves: if $G$ and $H$ are finitely generated quasi-isometric groups and $\mathcal{P}$ is a qi-characteristic collection of subgroups of $G$, then there is a collection of subgroups $\mathcal{Q}$ of $H$ such that $ (G, \mathcal{P})$ and $(H, \mathcal{Q})$ are quasi-isometric pairs. The second part of the article studies the number of filtered ends $\tilde e (G, P)$ of a pair of groups, a notion introduced by Bowditch, and provides an application of our main result: if $G$ and $H$ are quasi-isometric groups and $P\leq G$ is qi-characterstic, then there is $Q\leq H$ such that $\tilde e (G, P) = \tilde e (H, Q)$.

Locally Homogeneous Manifolds Defined by Lie Algebra of Infinitesimal Affine Transformations

This article deals with Lie algebra G of all infinitesimal affine transformations of the manifold M with an affine connection, its stationary subalgebra ℌ⊂G, the Lie group G corresponding to the algebra G, and its subgroup H⊂G corresponding to the subalgebra ℌ⊂G. We consider the center ℨ⊂G and the commutant [G,G] of algebra G. The following condition for the closedness of the subgroup H in the group G is proved. If ℌ∩ℨ+G;G=ℌ∩[G;G], then H is closed in G. To prove it, an arbitrary group G is considered as a group of transformations of the set of left cosets G/H, where H is an arbitrary subgroup that does not contain normal subgroups of the group G. Among these transformations, we consider right multiplications. The group of right multiplications coincides with the center of the group G. However, it can contain the right multiplication by element 𝒽¯, belonging to normalizator of subgroup H and not belonging to the center of a group G. In the case when G is in the Lie group, corresponding to the algebra G of all infinitesimal affine transformations of the affine space M and its subgroup H corresponding to its stationary subalgebra ℌ⊂G, we prove that such element 𝒽¯ exists if subgroup H is not closed in G. Moreover 𝒽¯ belongs to the closures H¯ of subgroup H in G and does not belong to commutant G,G of group G. It is also proved that H is closed in G if P+ℨ∩ℌ=P∩ℌ for any semisimple algebra P∈G for which P+ℜ=G.

REVISITING SOME FUZZY ALGEBRAIC STRUCTURES

Following our investigations on some fuzzy algebraic structures started in [6--8], and [9], in the present work, we revisit fuzzy groups and fuzzy ideals and introduce some new examples and then define the notion of fuzzy relation modulo a fuzzy subgroup and modulo a fuzzy ideal. As a consequence, we introduce the right and left cosets modulo a fuzzy relation. This work and the previously cited ones can be considered as a continuation of investigations initiated in [1--5]. Toward our investigation, we have in mind that by introducing these new definitions, the results that we can get should represent generalization of classical and commonly known concepts of algebra.

The Application of GAP Software in Constructing the Non-Normal Subgroup Graphs of Alternating Groups

A graph in group theory is constructed by using any elements of a group as a set of vertices. Some of the properties of a group are used to form the edges of the graph. A finite group can be represented in a graph by its subgroup structure. A subgroup H of a group G is a subset of G, where H itself is a group under the same operation as in G, whereas a subgroup H is said to be a normal subgroup if its left and right cosets coincide. The non-normal subgroup graph of a group G is defined as a directed graph with a vertex set G and two distinct elements x and y are adjacent if xy ∈ H. In this paper, the non-normal subgroups graph of alternating groups for order twelve is determined by using GAP software. The graphs are found to be a union of complete digraphs and directed graphs with the same pattern depending on the order of the non-normal subgroups.

A Generalization of Group-Graded Modules

In this article, we generalize the concept of group-graded modules by introducing the concept of G-weak graded R-modules, which are R-modules graded by a set G of left coset representatives, where R is a G-weak graded ring. Moreover, we prove some properties of these modules. Finally, results related to G-weak graded fields and their vector spaces are deduced. Many considerable examples are provided with more emphasis on the symmetric group S3 and the dihedral group D6, which gives the group of symmetries of a regular hexagon.

Function kernels and divisible groupoids

<abstract><p>In this paper, we introduce the notion of a function kernel which was motivated from the kernel in group theory, and we apply this notion to several algebraic structures, e.g., groups, groupoids, $ BCK $-algebras, semigroups, leftoids. Using the notions of left and right cosets in groupoids, we investigate some relations with function kernels. Moreover, the notion of an idenfunction in groupoids is introduced, which is a generalization of an identity axiom in algebras by functions, and we discuss it with function kernels.</p></abstract>

Fuzzy cosets in AG-groups

<abstract><p>In this paper, the notion of fuzzy AG-subgroups is further extended to introduce fuzzy cosets in AG-groups. It is worth mentioning that if $ A $ is any fuzzy AG-subgroup of $ G $, then $ \mu_{A}(xy) = \mu_{A}(yx) $ for all $ x, \, y\in G $, i.e. in AG-groups each fuzzy left coset is a fuzzy right coset and vice versa. Also, fuzzy coset in AG-groups could be empty contrary to fuzzy coset in group theory. However, the order of the nonempty fuzzy coset is the same as the index number $ [G:A] $. Moreover, the notions of fuzzy quotient AG-subgroup, fuzzy AG-subgroup of the quotient (factor) AG-subgroup, fuzzy homomorphism of AG-group and fuzzy Lagrange's theorem of finite AG-group is also introduced.</p></abstract>

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.

Quantitative structure of stable sets in arbitrary finite groups

We show that a k k -stable set in a finite group can be approximated, up to given error ϵ > 0 \epsilon >0 , by left cosets of a subgroup of index ϵ - O k ( 1 ) \epsilon ^{\text {-} O_k(1)} . This improves the bound in a similar result of Terry and Wolf on stable arithmetic regularity in finite abelian groups, and leads to a quantitative account of work of the author, Pillay, and Terry on stable sets in arbitrary finite groups. We also prove an analogous result for finite stable sets of small tripling in arbitrary groups, which provides a quantitative version of recent work by Martin-Pizarro, Palacín, and Wolf. Our proofs use results on VC-dimension, and a finitization of model-theoretic techniques from stable group theory.

Cardinal invariants of coset spaces

A topological space X is called a coset space if X is homeomorphic to a quotient space G/H of left cosets, for some closed subgroup H of a topological group G. In this paper, we investigate the cardinal invariants of coset spaces. We first show that if H is a closed neutral subgroup of a topological group G, then △(G/H)=ψ(G/H), w(G/H)=d(G/H)⋅χ(G/H) and w(G/H)=l(G/H)⋅χ(G/H).We also prove that if H is a closed subgroup of a feathered topological group G, then (1) w(G/H)=d(G/H)⋅χ(G/H) and w(G/H)=l(G/H)⋅χ(G/H); (2) the quotient space G/H is metrizable if and only if G/H is first-countable.At the end, we consider some applications of sp-networks in coset spaces. In particular, we show that if H is a closed neutral subgroup of a topological group G, then (1) spnw(G/H)=d(G/H)⋅spχ(G/H); (2) the quotient space G/H is metrizable if and only if G/H has countable sp-character.

Pythagorean Fuzzy N-Soft Groups

<p>We elaborate in this paper a new structure Pythagorean fuzzy<br />$N$-soft groups which is the generalization of intuitionistic fuzzy<br />soft group initiated by Karaaslan in 2013. In Pythagorean fuzzy<br />N-soft sets concepts of fuzzy sets, soft sets, N-soft sets, fuzzy<br />soft sets, intuitionistic fuzzy sets, intuitionistic fuzzy soft<br />sets, Pythagorean fuzzy sets, Pythagorean fuzzy soft sets are<br />generalized. We also talk about some elementary basic concepts and<br />operations on Pythagorean fuzzy N-soft sets with the assistance of<br />illusions. We additionally define three different sorts of<br />complements for Pythagorean fuzzy N-soft sets and examined a few<br />outcomes not hold in Pythagorean fuzzy N-soft sets complements as<br />they hold in crisp set hypothesis with the assistance of counter<br />examples. We further talked about {$(\alpha, \beta, \gamma)$-cut of<br />Pythagorean fuzzy N-soft set and their properties}. We likewise talk<br />about some essential properties of Pythagorean fuzzy N-soft groups<br />like groupoid, normal group, left and right cosets, $(\alpha, \beta,<br />\gamma)$-cut subgroups and some fundamental outcomes identified with<br />these terms. Pythagorean fuzzy N-soft sets is increasingly efficient<br />and adaptable model to manage uncertainties. The proposed models of<br />Pythagorean fuzzy N-soft groups can defeat a few disadvantages of<br />the existing statures.</p>

Near-rings on nearness approximation spaces

In this study, nearness near-ring, subnearness near-ring, nearness M-group and nearness ideal are introduced. By considering operations on the set of all near left weak cosets, nearness near-ring of all near left weak cosets and nearness near-ring homomorphism are also presented. Moreover, some properties of these structures are investigated.

Normal Bipolar Soft Subgroups

In this study, we introduce the concept of normal bipolar soft subgroup and investigate some properties of normal bipolar soft subgroups. We also define notions of bipolar soft left and right cosets of a bipolar soft group and the concept of conjugate bipolar soft subgroup, and obtain some of their properties.

Orthogonal Gyrodecompositions of Real Inner Product Gyrogroups

In this article, we prove an orthogonal decomposition theorem for real inner product gyrogroups, which unify some well-known gyrogroups in the literature: Einstein, Möbius, Proper Velocity, and Chen’s gyrogroups. This leads to the study of left (right) coset partition of a real inner product gyrogroup induced from a subgyrogroup that is a finite dimensional subspace. As a result, we obtain gyroprojectors onto the subgyrogroup and its orthogonal complement. We construct also quotient spaces and prove an associated isomorphism theorem. The left (right) cosets are characterized using gyrolines (cogyrolines) together with automorphisms of the subgyrogroup. With the algebraic structure of the decompositions, we study fiber bundles and sections inherited by the gyroprojectors. Finally, the general theory is exemplified for the aforementioned gyrogroups.