Non-abelian Groups Of Order Pq Research Articles

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.

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.

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.

Minimum product set sizes in nonabelian groups of order pq

Let G be a nonabelian group of order pq, where p and q are distinct odd primes. We analyze the minimum product set cardinality μ G ( r , s ) = min | A B | , where A and B range over all subsets of G of cardinalities r and s, respectively. In this paper, we completely determine μ G ( r , s ) in the case where G has order 3 p and conjecture that this result can be extended to all nonabelian groups of order pq. We also prove that for every nonabelian group of order pq there exist 1 ⩽ r , s ⩽ p q such that μ G ( r , s ) > μ Z / p q Z ( r , s ) .

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.

Improving the Erdős–Ginzburg–Ziv theorem for some non-abelian groups

Let G be a group of order m. Define s ( G ) to be the smallest value of t such that out of any t elements in G, there are m with product 1. The Erdős–Ginzburg–Ziv theorem gives the upper bound s ( G ) ⩽ 2 m − 1 , and a lower bound is given by s ( G ) ⩾ D ( G ) + m − 1 , where D ( G ) is Davenport's constant. A conjecture by Zhuang and Gao [J.J. Zhuang, W.D. Gao, Erdős–Ginzburg–Ziv theorem for dihedral groups of large prime index, European J. Combin. 26 (2005) 1053–1059] asserts that s ( G ) = D ( G ) + m − 1 , and Gao [W.D. Gao, A combinatorial problem on finite abelian groups, J. Number Theory 58 (1996) 100–103] has proven this for all abelian G. In this paper we verify the conjecture for a few classes of non-abelian groups: dihedral and dicyclic groups, and all non-abelian groups of order pq for p and q prime.

Some applications of pq-groups in graph theory

We describe some new applications of nonabelian pq-groups to construction problems in Graph Theory. The constructions include the smallest known trivalent graph of girth 17, the smallest known regular graphs of girth five for several degrees, along with four edge colorings of complete graphs that improve lower bounds on classical Ramsey numbers. For primes p and q such that p < q and q ≡ 1 (mod p), there exists a nonabelian group of order pq, which is a semidirect product of Zq by Zp [3]. In this note, we use these groups to construct several graphs and graph colorings. Let G be a nonabelian pq-group and let a and b be elements of orders p and q, respectively. The subgroup generated by b is normal in G and a−1ba = b where r ≡ 1 (mod q). So the elements of G can be represented in the form ab , for 0 ≤ i < p and 0 ≤ j < q. Group multiplication can be defined by ab ∗ ab = ab s+t. In what follows, it will be convenient to refer to elements of a pq-group using the notation (i, j) instead of ab . 1 Ramsey Graphs Recall that an (s, t)-coloring of the complete graph Kn is a 2-coloring of its edges such that there is neither a complete subgraph of order s, all of whose edges are color 1, nor a complete subgraph of order t, all of whose edges are color 2. The Ramsey number R(s, t) is the minimum n such that no (s, t) coloring on Kn exists. Our first application of pq-groups gives us a new lower bound for R(4, 8). For this we use a Cayley coloring on the nonabelian group G of order 55, where we take p = 5, q = 11, and r = 3. Note that in a Cayley coloring, we assign

Complete latin squares of order pn exist for odd primes p and n>2

A sequencing in a finite group G is a list of all elements of G such that the partial products of the list are all distinct. The existence of a sequencing in the nonabelian group of order p n which contains an element of order p n−1 , where p is an odd prime and n>2, is demonstrated. It follows that complete latin squares exist for these orders. A sufficient condition for the existence of a sequencing in the nonabelian group of order pq is also given, where p< q are odd primes with q=2 h+1 for some integer h.

On r*‐sequenceability and symmetric harmoniousness of groups. II

AbstractIn this paper we investigate symmetric harmoniousness of groups and connections of this concept to the R*‐sequenceability of groups. We prove that, under suitable assumptions, the direct product of a symmetric harmonious group with a group that is R*‐sequenceable is R*‐sequenceable; we discuss the symmetric harmoniousness of abelian and of nilpotent groups; we also prove that, for a fixed odd prime p, all but possibly finitely many of the nonabelian groups of order pq (q prime, q ≡ 1 (mod p)) are symmetric harmonious. © 1995 John Wiley &amp; Sons, Inc.

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

On the sequenceability of non-abelian groups of order pq

Let p be an odd prime which has 2 as a primitive root and let q be another odd prime of the form q=2 ph+1, then the non-abelian group of order pq (for existence of which the latter requirement is the well-known necessary and sufficient condition) is sequenceable.

Adders for the patterned starter in some non-abelian groups

Starters with adders in abelian groups of odd order have been used extensively in the construction and study of Room squares. It is possible to define these concepts in non-abelian groups, with similar applicability to Room squares. Examples are given of adders for the “patterned starter” in the non-abelian groups of order pq, p and q primes larger than 3, p ≡ q ≡ 3 (mod 4), q ≡ 1 (mod p).

Non-Abelian Groups of Order pq

Non-Abelian Groups of Order pq