Nontrivial difference sets in groups of order a power of 2 are part of the family of difference sets called Hadamard difference sets. In the abelian case, a group of order 22 tq2 has a difference set if and only if the exponent of the group is less tq2 Ž than or equal to 2 . In a previous work R. A. Liebler and K. W. Smith, in ‘‘Coding Theory, Design Theory, Group Theory: Proc. of the Marshall ...

In this paper we study the existence of free nonabelian subgroups in noncentral permutable general skew linear groups and locally finite group algebras.

Let be a connected Cayley graph of group G, then Γ is called normal if the right regular representation of G is a normal subgroup of , the full automorphism group of Γ. For the case where G is a finite nonabelian simple group and Γ is symmetric cubic Cayley graph, Caiheng Li and Shangjin Xu proved that Γ is normal with only two exceptions. Since then, the normality of nonsymmetric cubic Cayley ...

We study a system of nonabelian anyons in the lowest Landau level of a strong magnetic field. Using diagrammatic techniques, we prove that the virial coefficients do not depend on the statistics parameter. This is true for all representations of all nonabelian groups for the statistics of the particles and relies solely on the fact that the effective statistical interaction is a traceless opera...

A detailed study is presented of classical field configurations describing nonabelian vortices in two spatial dimensions, when a global SO(3) symmetry is spontaneously broken to a discrete group IK isomorphic to the group of integers mod 4. The vortices in this model are characterized by the nonabelian fundamental group π1(SO(3)/IK) =Q8, which is isomorphic to the group of quaternions. We prese...

Let G be a finite group and let x ∈ G. Define the order subset of G determined by x to be the set of all elements in G that have the same order as x. A group G is said to have perfect order subsets if the number of elements in each order subset of G is a divisor of |G|. In this article we prove a theorem for a class of nonabelian groups, which is analogous to Theorem 4 in [2]. We then prove tha...

Nonabelian group-based public key cryptography is a relatively new and exciting research field. Rapidly increasing computing power and the futurity quantum computers [52] that have since led to, the security of public key cryptosystems in use today, will be questioned. Research in new cryptographic methods is also imperative. Research on nonabelian group-based cryptosystems will become one of c...

Let w1 and w2 be nontrivial words in free groups Fn1 and Fn2 , respectively. We prove that, for all sufficiently large finite nonabelian simple groups G, there exist subsets C1 ⊆ w1(G) and C2 ⊆ w2(G) such that |Ci | = O(|G|1/2 log |G|) and C1C2 = G. In particular, ifw is any nontrivial word and G is a sufficiently large finite nonabelian simple group, then w(G) contains a thin base of order 2. ...

In this paper, we complete the classification of those finite 3groups G whose integral group rings have the multiplicative Jordan decomposition property. If G is abelian, then it is clear that Z[G] satisfies MJD. In the nonabelian case, we show that Z[G] satisfies MJD if and only if G is one of the two nonabelian groups of order 33 = 27.

In this paper, we essentially classify those finite 3-groups G having integral group rings with the multiplicative Jordan decomposition property. If G is abelian, then it is clear that Z[G] satisfies MJD. Thus, we are only concerned with the nonabelian case. Here we show that Z[G] has the MJD property for the two nonabelian groups of order 33. Furthermore, we show that there are at most three o...