Groups Of Order 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

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.

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 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.

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 NORMALITY OF SOME CAYLEY DIGRAPHS WITH VALENCY 2

We call a Cayley digraph 􀀀=Cay(G; S) normal for G if R(G), the rightregular representation of G, is a normal subgroup of the full automorphism groupAut(􀀀) of 􀀀. In this paper we determine the normality of Cayley digraphs of valency2 on the groups of order pq and also on non-abelian nite groups G such that everyproper subgroup of G is abelian.DOI : http://dx.doi.org/10.22342/jims.17.2.3.67-72

Graphs on which a group of order pq acts edge-transitively

Let Γ be a finite simple undirected graph with no isolated vertices. Let p, q be prime numbers with p ≥ q. We complete the classification of the graphs on which a group of order pq acts edge-transitively. The results are the following. If Aut(Γ) contains a subgroup G of order pq that acts edge-transitively on Γ, then Γ is one of the following graphs: (1) pK1,1; (2) pqK1,1; (3) pKq,1; (4) qKp,1 (p > q); (5) pCq (q > 2); (6) qCp (p > q); (7) Cp (p > q = 2); (8) Cpq; (9) (Zp,C) where C = {±rµ |µ∈ Zq{ with q > 2, q|(p − 1) and r ≢ 1 ≡ rq (mod p); (10) Kp,1 (p > q); (11) a double Cayley graph B(G,C) with C = {1 − rµ | µ ∈ Zq} and r ≢ 1 ≡ rq (mod p); (12) Kpq,1; or (13) Kp,q.

A nuclear construction of loops with small inner mapping groups

We construct an infinite class of left conjugacy closed loopsQ such that the inner mapping group InnQ is a nonabelian group of orderpq. The constructions are done in a broader context of loopsQ =N λ N ρ such thatN λ =N μ ⊴Q and InnQ can be embedded into the holomorph of a cyclic group.

Polynomial clones on groups of order pq

For two distinct primes p, q, we describe those clones on a set of size pq that contain a given group operation and all constant operations. We show that each such clone is determined by congruences and commutator relations. Thus we obtain that there is only a finite number of such clones on a fixed set.

On Groups of Order pq

On Groups of Order pq

On Groups of Order pq

The characteristic that distinguishes finite group theory from other branches of algebra is the nature of many important theorems. Indeed, proofs in finite group theory often amount to counting; Lagrange's theorem and the Sylow theorems are quintessential examples of the genre. Another highly appealing arithmetical result that is typically presented as an application of the Sylow theorems is that a group of order pq, where p and q are primes, q < p and q does not divide p 1, is cyclic (see, for example, [1, p. 245], [2, p. 358] or [3, p. 495]). The standard proof of this statement is as follows. By the Sylow theorems there are subgroups H and K of orders p and q that are normal. From the normality of H and K and the fact that H n K = {e}, it follows that elements of H and K commute with each other (see [2, p. 358]). Thus, if IhI = p and Ik I = q, then I hk I = and the group is cyclic. Since the Sylow theorems occur late in many undergraduate abstract algebra textbooks, most students who take only one semester of algebra will not encounter them. In this note we offer a proof of the pq result that does not use the Sylow theorems or normality. It can be given as an application of cosets and Lagrange's theorem. Our argument is an excellent illustration of sophisticated counting. The proof utilizes the following easy-to-prove facts.

Cyclotomic units over finite fields

Let ζ be a primitivesp-th root of unity for a primep>2, and consider the group Ω(ζ) of cyclotomic units in the ringR(ζ)=ℒ[ζ+ζ-1]. This paper deals with the image of Ω(ζ) in the unit group ofR(ζ)/qR(ζ), whereq is a prime ≠p. In particular, it obtains criteria for this image to be essentially everything, and a lower bound on the density of primesp (withq fixed) for which it cannot be. These results have a direct bearing on previous work about units in integral group rings for cyclic groups of orderpq.

Subplanes of a tactical decomposition and singer groups of a projective plane

Sufficient and necessary conditions have been obtained for the following: (1) the substructure formed by a member of the partition of points and a member of the partition of lines to be a subplane; (2) the centralizer of a multiplier to be a Baer subplane. We establish the cyclicity of a Sylow 3-subgroup of the multiplier group of an abelian Singer group of square planar order. Sufficient conditions for the existence of a Type II divisor of a Singer group are given. For a Singer group of orderpq, p<q, we prove that if the order of the multiplier group is divisible byp, then the plane will admit a cyclic Singer group.

On the generators of subgroups of unit groups of group rings

In this paper we find the generators of a subgroup of finite index in the unit group of the integral group ring of the metacyclic group of orderpq given byG=(a,x:a p=1=x q ,xax −1=a f ), wherep is an odd prime,q>2 a divisor ofp-1, andf belongs to the exponentq modulop.

Modules over the integral group ring of a nonabelian group of order 𝑝𝑞

Modules over the integral group ring of a nonabelian group of order 𝑝𝑞

Integral Group Rings of Some p-Groups

1. Introduction. The group of units, , of the integral group ring of a finite non-abelian group G is difficult to determine. For the symmetric group of order 6 and the dihedral group of order 8 this was done by Hughes-Pearson [3] and Polcino Milies [5] respectively. Allen and Hobby [1] have computed , where A4 is the alternating group on 4 letters. Recently, Passman-Smith [6] gave a nice characterization of where D2p is the dihedral group of order 2p and p is an odd prime. In an earlier paper [2] Galovich-Reiner-Ullom computed when G is a metacyclic group of order pq with p a prime and q a divisor of (p – 1). In this note, using the fibre product decomposition as in [2], we give a description of the units of the integral group rings of the two noncommutative groups of order p3, p an odd prime. In fact, for these groups we describe the components of ZG in the Wedderburn decomposition of QG.

Groups of Order pq m with Elementary Abelian Sylow q-Subgroups

Groups of Order pq m with Elementary Abelian Sylow q-Subgroups

Groups of order 𝑝𝑞^{𝑚} with elementary abelian Sylow 𝑞-subgroups

Characterization and number of the groups of order p q m p{q^m} with elementary Abelian Sylow q-subgroups. Old methods of O. Hölder and A. E. Western are simplified and generalized.

Actions of Groups of Order pq

Actions of Groups of Order pq