Groups Of Pq Research Articles

A Classification of 4-Degree Tri-Cayley Graphs Over a Group of Order 𝑝q

A Classification of 4-Degree Tri-Cayley Graphs Over a Group of Order 𝑝q

Acentralizers of some Finite Groups

Let G be a finite group. The acentralizer of an automorphism α of G, is the subgroup of fixed points of α, i.e., CG(α) = {g ∈ G∣α(g) = g}. In this paper we determine the acentralizers of the dihedral group of order 2n, the dicyclic group of order 4n and the symmetric group on n letters. As a result we see that if n ≥ 3, then the number of acentralizers of the dihedral group and the dicyclic group of order 4n are equal. Also we determine the acentralizers of groups of orders pq and pqr, where p, q and r are distinct primes.

Degree and distance based topological descriptors of power graphs of finite non-abelian groups

A topological descriptor is a numerical value derived from the molecular structure that encapsulates the most important structural characteristics of the molecule under consideration. Fundamentally, it involves assigning an algebraic value to the composition of chemicals while developing a relationship between this value and several physical properties, like biological activity, and chemical reactivity. This article examines multiple kinds of degree and eccentricity-based topological indices for power graphs of various finite groups. We calculate the Wiener index and its reciprocal, atom-bond connectivity index and its fourth version, the Schultz index, the geometric–arithmetic and harmonic indices, and finally determine the general Randić and Harary indices of power graphs of finite cyclic and non-cyclic groups of order pq, dihedral, and generalized quaternion groups, where p,q(q≥p) are distinct primes.

Indecomposable solutions of the Yang-Baxter equation with permutation group of sizes pq and p 2 q

In this paper we study the problem of the classification of indecomposable solutions of the Yang-Baxter equation. Using a scheme proposed by Bachiller, Cedó, and Jespers, and recent advances in the classification of braces we classify all indecomposable solutions with some particular permutation groups. We do this for all groups of size pq, all abelian groups of size p 2 q and all dihedral groups of size p 2 q.

On the intersection density of primitive groups of degree a product of two odd primes

A subset F of a finite transitive group G≤Sym(Ω) is intersecting if for any g,h∈F there exists ω∈Ω such that ωg=ωh. The intersection densityρ(G) of G is the maximum of {|F||Gω||F⊂G is intersecting}, where Gω is the stabilizer of ω in G. In this paper, it is proved that if G is an imprimitive group of degree pq, where p and q are distinct odd primes, with at least two systems of imprimitivity then ρ(G)=1. Moreover, if G is primitive of degree pq, where p and q are distinct odd primes, then it is proved that ρ(G)=1, whenever the socle of G admits an imprimitive subgroup.

Hosoya Polynomials of Power Graphs of Certain Finite Groups.

Assume that is a finite group. The power graph of is a graph in which is its node set, where two different elements are connected by an edge whenever one of them is a power of the other. A topological index is a number generated from a molecular structure that indicates important structural properties of the proposed molecule. Indeed, it is a numerical quantity connected with the chemical composition that is used to correlate chemical structures with various physical characteristics, chemical reactivity, and biological activity. This information is important for identifying well-known chemical descriptors based on distance dependence. In this paper, we study Hosoya properties, such as the Hosoya polynomial and the reciprocal status Hosoya polynomial of power graphs of various finite cyclic and non-cyclic groups of order and , where and are prime numbers.

On intersection density of transitive groups of degree a product of two odd primes

Two elements g and h of a permutation group G acting on a set V are said to be intersecting if g(v)=h(v) for some v∈V. More generally, a subset F of G is an intersecting set if every pair of elements of F is intersecting. The intersection densityρ(G) of a transitive permutation group G is the maximum value of the quotient |F|/|Gv| where Gv is the stabilizer of v∈V and F runs over all intersecting sets in G. Intersection densities of transitive groups of degree pq, where p>q are odd primes, is considered. In particular, the conjecture that the intersection density of every such group is equal to 1 (posed in Meagher et al. (2021) [15]) is disproved by constructing a family of imprimitive permutation groups of degree pq (with blocks of size q), where p=(qk−1)/(q−1), whose intersection density is equal to q. The construction depends heavily on certain equidistant cyclic codes [p,k]q over the field Fq whose codewords have Hamming weight strictly smaller than p.

Solving a fixed number of equations over finite groups

We investigate the complexity of solving systems of polynomial equations over finite groups. In 1999 Goldmann and Russell showed mathrm {NP}-completeness of this problem for non-Abelian groups. We show that the problem can become tractable for some non-Abelian groups if we fix the number of equations. Recently, Földvári and Horváth showed that a single equation over groups which are semidirect products of a p-group with an Abelian group can be solved in polynomial time. We generalize this result and show that the same is true for systems with a fixed number of equations. This shows that for all groups for which the complexity of solving one equation has been proved to be in mathrm {P} so far, solving a fixed number of equations is also in mathrm {P}. Using the collecting procedure presented by Horváth and Szabó in 2006, we furthermore present a faster algorithm to solve systems of equations over groups of order pq.

Bol loops and Bruck loops of order pq up to isotopism

Let p>q be odd primes. We classify Bol loops and Bruck loops of order pq up to isotopism. When q does not divide p2−1, the only Bol loop (and hence the only Bruck loop) of order pq is the cyclic group of order pq. When q divides p2−1, there are precisely ⌊(p−1+4q)(2q)−1⌋ Bol loops of order pq up to isotopism, including a unique nonassociative Bruck loop of order pq.

Bol loops and Bruck loops of order pq

Right Bol loops are loops satisfying the identity ((zx)y)x=z((xy)x), and right Bruck loops are right Bol loops satisfying the identity (xy)−1=x−1y−1. Let p and q be odd primes such that p>q. Advancing the research program of Niederreiter and Robinson from 1981, we classify right Bol loops of order pq. When q does not divide p2−1, the only right Bol loop of order pq is the cyclic group of order pq. When q divides p2−1, there are precisely (p−q+4)/2 right Bol loops of order pq up to isomorphism, including a unique nonassociative right Bruck loop Bp,q of order pq.Let Q be a nonassociative right Bol loop of order pq. We prove that the right nucleus of Q is trivial, the left nucleus of Q is normal and is equal to the unique subloop of order p in Q, and the right multiplication group of Q has order p2q or p3q. When Q=Bp,q, the right multiplication group of Q is isomorphic to the semidirect product of Zp×Zp with Zq. Finally, we offer computational results as to the number of right Bol loops of order pq up to isotopy.

On Finite Nonabelian Loop Capable Groups

We say that groups, which are isomorphic to inner mapping groups of finite loops, are loop capable. Let p and q be distinct prime numbers, S a nonabelian group of order pq, and C a finite nontrivial cyclic group such that gcd (|S|, |C|) = 1. We show that the group S × C is not loop capable.

The effect of Ulinastatin on autophagy and apoptosis in the acute paraquat poisoning rats lung cellular

Objective To investigate the effects of ulinastatin on autophagy and apoptosis of lung cells in rats with acute paraquat poisoning. Methods A total of 150 Wistar rats were randomly (random number) divided into three groups. The rats in control group had stomach lavaged once with 1 mL of normal saline followed by intraperitoneal injection of 1 mL normal saline twice a day. PQ poisoning model was produced by stomach lavaged once with 1 mL of 40 mg/kg PQ solution followed by intraperitoneal injection of 1 mL normal saline once a day. In PQ + ulinastatin (PU) group, UTI in dose of 12 000 U/kg was intraperitoneally injected in rats twice a day. The lung tissue was obtained on the 7th day after modeling, and the histopathological changes were observed under microscope after hematoxylin and eosin (HE) staining. The positive expressions of autophagy-related LC3 protein LC3 and Bcl-2 pretein in lung tissue were observed after immunohistochemistry staining, and the levels of LC3、Bax、Bcl-2 proteins were determined by Western blot. Results HE staining Results showed: it was observed from the PQ poisoning group that the abnormal cellular structure, enlargement in the pulmonary alveoli, leaking a lot of inflammatory cells, increased thickness of the alveoli wall and bleeding in the local area of lung tissue. Compared with the PQ poisoning group, the above changes in ulinastatin groups were relieved . Western blot Results showed: compared with the control group, the protein expressions of LC3-A/B were significantly increased in PQ poisoning group [LC3-A/B expression (A scale) : 0.22 ±0.05 vs. 0.14±0.03, F =22.48, P <0.01 ]. compared with PQ group, the expression of LC3 A/B obviously increased in the group of PU [LC3-A/B expression (A scale) : 0.36 ±0.08 vs. 0.22 ±0.05, F =22.78, P <0.01] . compared with Con group, the expression of Bcl-2/Bax obviously decreased in the group of PQ [Bcl-2/Bax expression (A scale) , 0.11 ±0.04 vs. 0.83 ± 0.09, F =154.43, P <0.01 ]. Compared with PQ poisoning group, the protein expressions of Bcl-2/Bax were obviously increased in PU groups [Bcl-2/Bax expression (A scale) : (0.63 ± 018) vs. (0.11 ± 0.04) , F =154.43, P <0.01]. Immunohistochemistry result: compared with Con group, the expression of LC3 and Bcl-2 obviously decreased in the group of PQ [LC3expression (A scale) : (78.34±10.71) vs. (117.58±15.26) , F=31.63, P<0.01) (Bcl-2 expression (A scale): (62.54±9.74) vs. (130.52 ±9.86, F =118.44, P <0.01) . Compared with PQ poisoning group, the protein expressions of LC3 and Bcl-2 were obviously increased in PU groups [LC3expression (A scale) : (162.58 ±25.76) vs. (78.34 ± 10.71) , F=31.63, P<0.01]; [Bcl-2 expression (A scale) : (145.56 ±10.26) vs. (62.54±9.74) , F= 118.44, P <0.01]. Conclusions The endoplasmic reticulum stress -autophagy is activated in the lung cells of rats with acute PQ poisoning. UTI can adjust endoplasmic reticulum stress, increased the expression of Bcl-2 and enhance the proportion of Bcl-2/Bax to protect the lungs of rats from acute PQ poisoning. Key words: Paraquat; Poisoning; Rat; Ulinastatin; Autophagy; Apoptosis; LC3; Bcl-2; Bax

The Orbit Graph of Finite p-Groups and Groups of Order pq

In this paper G denotes a finite p-group and Γ denotes a simple undirected graph. The orbit graph is a graph whose vertices are non-central orbit under group action of G on a set. Two vertices v 1 and v 2 are adjacent in the graph if v 1 g = v 2 where v 1 ,v 2 ∈Ω and g∈G. In this paper, the orbit graph of some finite p-groups and group of order pq, where p and q are relatively prime, is found. The orbit graph is determined for the group in the case that a group acts regularly on itself, acts on itself by conjugation, and acts on a set. Besides, some graph properties are found.

Harmonic numbers and finite groups

Given a finite group G , let {\tau} (G) be the number of normal subgroups of G and {\sigma} (G) be the sum of the orders of the normal subgroups of G . The group G is said to be harmonic if H(G):=\vert G\vert {\tau} (G)/{\sigma} (G) is an integer. In this paper, all finite groups for which 1 \leq H(G) \leq 2 have been characterized. Harmonic groups of order pq and of order pqr , where p&lt; q&lt; r are primes, are also classified. Moreover, it has been shown that if G is harmonic and G \not \cong C_ 6 , then {\tau} (G) \geq 6 .

Finite groups with a given absolute central factor group

Let G be a group and L(G) be the absolute center of G, that is, the set of all elements of G fixed by all automorphisms of G. In this paper, we classify all finite groups G whose absolute central factors are isomorphic to a cyclic group, \({\mathbb{Z}_p \times \mathbb{Z}_p}\) , D8, Q8, or a non-abelian group of order pq for some distinct primes p and q.

Non-CLT groups of order pq 3

Abstract In this note we give a characterization of finite groups of order pq 3 (p, q primes) that fail to satisfy the Converse of Lagrange’s Theorem.

On the Symmetry Classes of Tensors Associated with Certain Frobenius Groups

In this paper, we study the symmetry classes of tensors associated with some Frobenius groups of order pq, where q|p-1as a subgroups of the full symmetric group on p letters. We calculate the dimension of the symmetry classes of tensor associated with some Frobenius groups and some irreducible complex characters and we obtain two useful corollary with an example.

Minimum product sets sizes in nonabelian groups

Given a group G and integers r and s , let μ G ( r , s ) be the minimum cardinality of the product set AB , where A and B are subsets of G of cardinality r and s , respectively. We compute μ G for all nonabelian groups of order pq , where p and q are distinct odd primes, thus proving a conjecture of Deckelbaum. In addition, we apply a theorem of Eliahou and Kervaire to compute μ G for all finite nilpotent groups.

HIGHER INDICATORS FOR SOME GROUPS AND THEIR DOUBLES

In this paper we explicitly determine all indicators for groups isomorphic to the semidirect product of two cyclic groups by an automorphism of prime order, as well as the generalized quaternion groups. We then compute the indicators for the Drinfel'd doubles of these groups. This first family of groups include the dihedral groups, the non-abelian groups of order pq, and the semidihedral groups. We find that the indicators are all integers, with negative integers being possible in the first family only under certain specific conditions.

The η invariant of the Atiyah–Patodi–Singer operator on compact flat manifolds

Let \({\mathcal{D}}\) be the boundary operator defined by Atiyah, Patodi and Singer, acting on smooth even forms of a compact orientable Riemannian manifold M. In continuation of our previous study, we deal with the problem of computing explicitly the η invariant η = η(M) for any orientable compact flat manifold M. After giving an explicit expression for η(s) in the case of cyclic holonomy group, we obtain a combinatorial formula that reduces the computation to the cyclic case. We illustrate the method by determining η(0) for several infinite families, some of them having non-abelian holonomy groups. For cyclic groups of odd prime order p ≥ 7, η(s) can be expressed as a multiple of Lχ(s), an L-function associated to a quadratic character mod p, while η(0) is a (non-zero) integral multiple of the class number h−p of the number field \({\mathbb Q(\sqrt {-p})}\) . In the case of metacyclic groups of odd order pq, with p, q primes, we show that η(0) is a rational multiple of h−p.