The complete generalized cycle G(d, n) is the digraph which has Zn × Zd as the vertex set and every vertex (i, x) is adjacent to the d vertices (i + 1, y) with y ∈ Zd . As a main result, we give a necessary and sufficient condition for the iterated line digraph G(d, n, k) = Lk−1G(d, n), with d a prime number, to be a Cayley digraph in terms of the existence of a group0d of order d and a subgrou...

Arasu, K.T. and A. Pott, Cyclic affine planes and Paley difference sets, Discrete Mathematics 106/107 (1992) 19-23. The existence of a cyclic affine plane implies the existence of a Paley type difference set. We use the existence of this difference set to give the following condition on the existence of cyclic affine planes of order n: If n 8 mod 16 then n 1 must be a prime. We discuss the stru...

This paper gives a complete classification of the finite groups that contain a strongly closed p-subgroup for p any prime.

a normal subgroup $n$ of a group $g$ is said to be an‎ ‎$emph{omissible}$ subgroup of $g$ if it has the following property‎: ‎whenever $xleq g$ is such that $g=xn$‎, ‎then $g=x$‎. ‎in this note we construct various groups $g$‎, ‎each of which has an omissible subgroup $nneq 1$ such that $g/ncong sl_2(k)$ where $k$ is a field of positive characteristic‎.

For each odd prime p, we conjecture the distribution of the p-torsion subgroup of K2n(OF ) as F ranges over real quadratic fields, or over imaginary quadratic fields. We then prove that the average size of the 3-torsion subgroup of K2n(OF ) is as predicted by this conjecture.

It is shown that in the units of augmentation one of an integral group ring ZG of a finite group G, a noncyclic subgroup of order p, for some odd prime p, exists only if such a subgroup exists in G. The corresponding statement for p = 2 holds by the Brauer–Suzuki theorem, as recently observed by W. Kimmerle.

For a natural number n, let λ(n) denote the order of the largest cyclic subgroup of (Z/nZ). For a given integer a, let Na(x) denote the number of n ≤ x coprime to a for which a has order λ(n) in (Z/nZ). Let R(n) denote the number of elements of (Z/nZ) with order λ(n). It is natural to compare Na(x) with ∑ n≤x R(n)/n. In this paper we show that the average of Na(x) for 1≤ a ≤ y is indeed asympto...

We provide the first explicit construction of genus 2 curves over finite fields whose Jacobians are ordinary, have large prime-order subgroups, and have small embedding degree. Our algorithm works for arbitrary embedding degrees k and prime subgroup orders r. The resulting abelian surfaces are defined over prime fields Fq with q ≈ r. We also provide an algorithm for constructing genus 2 curves ...

‎suppose $n$ is a fixed positive integer‎. ‎we introduce the relative n-th non-commuting graph $gamma^{n} _{h,g}$‎, ‎associated to the non-abelian subgroup $h$ of group $g$‎. ‎the vertex set is $gsetminus c^n_{h,g}$ in which $c^n_{h,g} = {xin g‎ : ‎[x,y^{n}]=1 mbox{~and~} [x^{n},y]=1mbox{~for~all~} yin h}$‎. ‎moreover‎, ‎${x,y}$ is an edge if $x$ or $y$ belong to $h$ and $xy^{n}eq y^{n}x$ or $x...

We provide the first explicit construction of genus 2 curves over finite fields whose Jacobians are ordinary, have large prime-order subgroups, and have small embedding degree. Our algorithm is modeled on the Cocks-Pinch method for constructing pairing-friendly elliptic curves [5], and works for arbitrary embedding degrees k and prime subgroup orders r. The resulting abelian surfaces are define...