scholarly journals Nonexistence of abelian difference sets: Lander’s conjecture for prime power orders

2003 ◽  
Vol 356 (11) ◽  
pp. 4343-4358 ◽  
Author(s):  
Ka Hin Leung ◽  
Siu Lun Ma ◽  
Bernhard Schmidt
Keyword(s):  
Prime Power ◽  
Difference Sets ◽  
Power Orders

Improved pairwise test suites for non‐prime‐power orders

IET Software ◽  
2018 ◽  
Vol 12 (3) ◽  
pp. 215-224 ◽  
Author(s):  
Himer Avila-George ◽  
Jose Torres-Jimenez ◽  
Idelfonso Izquierdo-Marquez
Keyword(s):  
Prime Power ◽  
Power Orders ◽  
Test Suites ◽  
Pairwise Test

GOLOMB RULERS AND DIFFERENCE SETS FOR SUCCINCT QUANTUM AUTOMATA

2003 ◽  
Vol 14 (05) ◽  
pp. 871-888 ◽  
Author(s):  
ALBERTO BERTONI ◽  
CARLO MEREGHETTI ◽  
BEATRICE PALANO
Keyword(s):  
Finite Automaton ◽  
Prime Power ◽  
Difference Sets ◽  
Minimal Size ◽  
Quantum Automata ◽  
Size Number ◽  
Minimal Difference ◽  
Golomb Rulers ◽  
Efficient Construction ◽  
Stochastic Event

Given a function p : N → [0,1] of period n, we study the minimal size (number of states) of a one-way quantum finite automaton (Iqfa) inducing the stochastic event ap + b, for real constants a>0, b≥0, a+b≤1. First of all, we relate the estimation of the minimal size to the problem of finding a minimal difference cover for a suitable subset of Zn. Then, by observing that the cardinality of a difference cover Δ for a set A ⊆ Zn, must satisfy [Formula: see text], we investigate the class of sets A admitting difference covers of cardinality exactly [Formula: see text]. We relate this problem with the efficient construction of Golomb rulers and difference sets. We design an algorithm which outputs each of the Golomb rulers (if any) of a given set in pseudo-polynomial time. As a consequence, we obtain an efficient algorithm that construct minimal difference covers for a non-trivial class of sets. Moreover, by using projective geometry arguments, we give an algorithm that, for any n=q2+q+1 with q prime power, constructs difference sets for Zn in quadratic time.

ON SETS OF PP-GENERATORS OF FINITE GROUPS

2014 ◽  
Vol 91 (2) ◽  
pp. 241-249 ◽  
Author(s):  
JAN KREMPA ◽  
AGNIESZKA STOCKA
Keyword(s):  
Finite Groups ◽  
Prime Power ◽  
Basis Property ◽  
Nilpotent Groups ◽  
Minimal Sets ◽  
Formula Group ◽  
Power Orders

AbstractThe classes of finite groups with minimal sets of generators of fixed cardinalities, named ${\mathcal{B}}$-groups, and groups with the basis property, in which every subgroup is a ${\mathcal{B}}$-group, contain only $p$-groups and some $\{p,q\}$-groups. Moreover, abelian ${\mathcal{B}}$-groups are exactly $p$-groups. If only generators of prime power orders are considered, then an analogue of property ${\mathcal{B}}$ is denoted by ${\mathcal{B}}_{pp}$ and an analogue of the basis property is called the pp-basis property. These classes are larger and contain all nilpotent groups and some cyclic $q$-extensions of $p$-groups. In this paper we characterise all finite groups with the pp-basis property as products of $p$-groups and precisely described $\{p,q\}$-groups.

scholarly journals Some classes of Hadamard matrices with constant diagonal

1972 ◽  
Vol 7 (2) ◽  
pp. 233-249 ◽  
Author(s):  
Jennifer Wallis ◽  
Albert Leon Whiteman
Keyword(s):  
Prime Power ◽  
Abelian Groups ◽  
Difference Sets ◽  
Hadamard Matrices ◽  
Incidence Matrices

The concepts of circulant and backcirculant matrices are generalized to obtain incidence matrices of subsets of finite additive abelian groups. These results are then used to show the existence of skew-Hadamard matrices of order 8(4f+1) when f is odd and 8f + 1 is a prime power. This shows the existence of skew-Hadamard matrices of orders 296, 592, 1184, 1640, 2280, 2368 which were previously unknown.A construction is given for regular symmetric Hadamard matrices with constant diagonal of order 4(2m + 1)2 when a symmetric conference matrix of order 4m + 2 exists and there are Szekeres difference sets, X and Y, of size m satisfying x є X ⇒ −xє X, y є Y ⇒ −y єY.

The prime power conjecture is true for $n < 2{,}000{,}000$

10.37236/1186 ◽  
1994 ◽  
Vol 1 (1) ◽  
Author(s):  
Daniel M. Gordon
Keyword(s):  
Prime Power ◽  
Necessary Conditions ◽  
Difference Sets ◽  
Cyclic Difference Sets

The Prime Power Conjecture (PPC) states that abelian planar difference sets of order $n$ exist only for $n$ a prime power. Evans and Mann verified this for cyclic difference sets for $n \leq 1600$. In this paper we verify the PPC for $n \leq 2{,}000{,}000$, using many necessary conditions on the group of multipliers.

scholarly journals Invariable generation with elements of coprime prime-power orders

2015 ◽  
Vol 423 ◽  
pp. 683-701 ◽  
Author(s):  
Eloisa Detomi ◽  
Andrea Lucchini
Keyword(s):  
Prime Power ◽  
Power Orders

scholarly journals Perfect difference sets

1964 ◽  
Vol 6 (4) ◽  
pp. 177-184 ◽  
Author(s):  
H. Halberstam ◽  
R. R. Laxton
Keyword(s):  
Prime Power ◽  
Difference Sets ◽  
Difference Set ◽  
Comprehensive Survey ◽  
Open Question

If the set K of r+1 distinct integers k0, k1 …, kr has the property that the (r+1)r differences ki–kj (0≦i, j≦r, i≠j) are distinct modulo r2+r+1, K is called a perfect difference set modr2+r+1. The existence of perfect difference sets seems intuitively improbable, at any rate for large r, but in 1938 J. Singer [1] proved that, whenever r is a prime power, say r = pn, a perfect difference set mod p2n+pn+1 exists. Since the appearance of Singer's paper several authors have succeeded in showing that for many kinds of number r perfect difference sets mod r2+r+1 do not exist; but it remains an open question whether perfect difference sets exist only when r is a prime power (for a comprehensive survey see [2]).

Finite Groups with Some Subgroups of Prime Power OrderS-Quasinormally Embedded

2004 ◽  
Vol 32 (5) ◽  
pp. 2019-2027 ◽  
Author(s):  
M. Asaad ◽  
A. A. Heliel ◽  
M. Ezzat Mohamed
Keyword(s):  
Finite Groups ◽  
Prime Power ◽  
Power Orders

scholarly journals The Influence of C- Z-permutable Subgroups on the Structure of Finite Groups

2018 ◽  
Vol 11 (1) ◽  
pp. 160
Author(s):  
Mohammed Mosa Al-shomrani ◽  
Abdlruhman A. Heliel
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Prime Power ◽  
Nonempty Subset ◽  
Permutable Subgroup ◽  
Sylow Subgroups ◽  
Permutable Subgroups ◽  
Complete Set ◽  
A Finite Group ◽  
Power Orders

Let Z be a complete set of Sylow subgroups of a finite group G, that is, for each prime p dividing the order of G, Z contains exactly one and only one Sylow p-subgroup of G, say Gp. Let C be a nonempty subset of G. A subgroup H of G is said to be C-Z-permutable (conjugateZ-permutable) subgroup of G if there exists some x ∈ C such that HxGp = GpHx, for all Gp ∈ Z. We investigate the structure of the finite group G under the assumption that certain subgroups of prime power orders of G are C-Z-permutable subgroups of G.

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