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.

On Two Properties of Shunkov Group

One of the interesting classes of mixed groups ( i.e. groups that can contain both elements of finite order and elements of infinite order) is the class of Shunkov groups. The group $G$ is called Shunkov group if for any finite subgroup $H$ of $G$ in the quotient group $N_G(H)/H$, any two conjugate elements of prime order generate a finite group. When studying the Shunkov group $G$, a situation often arises when it is necessary to move to the quotient group of the group $G$ by some of its normal subgroup $N$. In which cases is the resulting quotient group $G/N$ again a Shunkov group? The paper gives a positive answer to this question, provided that the normal subgroup $N$ is locally finite and the orders of elements of the subgroup $N$ are mutually simple with the orders of elements of the quotient group $G/N$. Let $ \mathfrak{X}$ be a set of groups. A group $G$ is saturated with groups from the set $ \mathfrak{X}$ if any finite subgroup of $G$ is contained in a subgroup of $ G$ that is isomorphic to some group of $\mathfrak{X}$ . If all elements of finite orders from the group $G$ are contained in a periodic subgroup of the group $G$, then it is called the periodic part of the group $G$ and is denoted by $T(G)$. It is proved that the Shunkov group saturated with finite linear and unitary groups of degree 3 over finite fields has a periodic part that is isomorphic to either a linear or unitary group of degree 3 on a suitable locally finite field. \end{abstracte}

On Periodic Shunkov’s Groups with Almost Layer-finite Normalizers of Finite Subgroups

Layer-finite groups first appeared in the work by S. N. Chernikov (1945). Almost layer-finite groups are extensions of layer-finite groups by finite groups. The class of almost layer-finite groups is wider than the class of layer-finite groups; it includes all Chernikov groups, while it is easy to give examples of Chernikov groups that are not layer-finite. The author develops the direction of characterizing well-known and well-studied classes of groups in other classes of groups with some additional (rather weak) finiteness conditions. A Shunkov group is a group 𝐺 in which for any of its finite subgroups 𝐾 in the quotient group <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msub><mi>N</mi><mi>G</mi></msub><mo>(</mo><mi>K</mi><mo>)</mo></mrow><mi>K</mi></mfrac></math> any two conjugate elements of prime order generate a finite subgroup. In this paper, we prove the properties of periodic not almost layer-finite Shunkov groups with condition: the normalizer of any finite nontrivial subgroup is almost layer-finite. Earlier, these properties were proved in various articles of the author, as necessary, sometimes under some conditions, then under others (the minimality conditions for not almost layer-finite subgroups, the absence of second-order elements in the group, the presence of subgroups with certain properties in the group). At the same time, it was necessary to make remarks that this property is proved in almost the same way as in the previous work, but under different conditions. This eliminates the shortcomings in the proofs of many articles by the author, in which these properties are used without proof.

On Structural Properties of ξ -Complex Fuzzy Sets and Their Applications

Complex fuzzy sets are the novel extension of Zadeh’s fuzzy sets. In this paper, we comprise the introduction to the concept of ξ -complex fuzzy sets and proofs of their various set theoretical properties. We define the notion of α , δ -cut sets of ξ -complex fuzzy sets and justify the representation of an ξ -complex fuzzy set as a union of nested intervals of these cut sets. We also apply this newly defined concept to a physical situation in which one may judge the performance of the participants in a given task. In addition, we innovate the phenomena of ξ -complex fuzzy subgroups and investigate some of their fundamental algebraic attributes. Moreover, we utilize this notion to define level subgroups of these groups and prove the necessary and sufficient condition under which an ξ -complex fuzzy set is ξ -complex fuzzy subgroup. Furthermore, we extend the idea of ξ -complex fuzzy normal subgroup to define the quotient group of a group G by this particular ξ -complex fuzzy normal subgroup and establish an isomorphism between this quotient group and a quotient group of G by a specific normal subgroup G A ξ .

The Automorphism Group of Green Algebra of 9-Dimensional Taft Hopf Algebra

Let H3 be the 9-dimensional Taft Hopf algebra, let [Formula: see text] be the corresponding Green ring of H3, and let [Formula: see text] be the automorphism group of Green algebra [Formula: see text] over the real number field ℝ. We prove that the quotient group [Formula: see text] is isomorphic to the direct product of the dihedral group of order 12 and the cyclic group of order 2, where T1 is the isomorphism class which contains the identity map and is isomorphic to a group [Formula: see text] with multiplication given by [Formula: see text].

The group of self-homotopy equivalences of An2{A_{n}^{2}}-polyhedra

Let X be a finite type {A_{n}^{2}}-polyhedron, {n\geq 2}. In this paper, we study the quotient group {\mathcal{E}(X)/\mathcal{E}_{*}(X)}, where {\mathcal{E}(X)} is the group of self-homotopy equivalences of X and {\mathcal{E}_{*}(X)} the subgroup of self-homotopy equivalences inducing the identity on the homology groups of X. We show that not every group can be realised as {\mathcal{E}(X)} or {\mathcal{E}(X)/\mathcal{E}_{*}(X)} for X an {A_{n}^{2}}-polyhedron, {n\geq 3}, and specific results are obtained for {n=2}.

The Tubby Torus as a Quotient Group

Let E be any metrizable nuclear locally convex space and E ^ the Pontryagin dual group of E. Then the topological group E ^ has the tubby torus (that is, the countably infinite product of copies of the circle group) as a quotient group if and only if E does not have the weak topology. This extends results in the literature related to the Banach–Mazur separable quotient problem.

Students' Thinking Process in Solving Mathematical Problems in Build Flat Side Spaces of Material Reviewed from Adversity Quotient

Mathematical achievement in problem solving is not yet as expected. This is due to strategies that have not accommodated the situation of students. Adversity quontient is the state of the student who needs to get the attention of the teacher so that the learning strategy is expected to be appropriate. This research aims to describe the thinking process of MTs Darul Hikmah students in solving mathematical problems in build side spaces of material reviewed from adversity quotient. This research is in the form of qualitative. The subject selection was done by giving questionnaires to class VIII MTsS Darul Hikmah students and obtained by each one person from each adversity quotient group. The subjects in this study amounted to two students. Data collection is done through tests and interviews, while data analysis uses qualitative data analysis techniques, namely data reduction, data display and conclusions from each adversity quotient group. The results showed that no class VIII students entered the quitters group and the results of the study also showed that class VIII students consisted of climpers and campers. The process of thinking of MTsS Darul Hikmah students based on adversity quotient varied: students in the climbers category namely 3 of 53 students had conceptual thinking processes, while students in the campers category were 27 of 53 students did not have conceptual, semiceptual, or computational thinking processes.