Representations of the cyclic groups of prime order 𝑝 over residue classes π‘šπ‘œπ‘‘π‘^{𝑠}

1968 β—½  
pp. 241-256 β—½  
Author(s):  
V. S. Drobotenko β—½  
È. S. Drobotenko β—½  
Z. P. Žilinskaja β—½  
E. Ja. Pogoriljak
Keyword(s):  
Prime Order β—½  
Cyclic Groups β—½  
Residue Classes

scholarly journals Atomic Latin Squares based on Cyclotomic Orthomorphisms

10.37236/1919 β—½  
2005 β—½  
Vol 12 (1) β—½  
Author(s):  
Ian M. Wanless
Keyword(s):  
Finite Fields β—½  
Complete Graph β—½  
Prime Order β—½  
Automorphism Groups β—½  
Latin Square β—½  
Latin Squares β—½  
Established Method β—½  
Cyclic Groups β—½  
Complete Bipartite Graphs β—½  
First Time

Atomic latin squares have indivisible structure which mimics that of the cyclic groups of prime order. They are related to perfect $1$-factorisations of complete bipartite graphs. Only one example of an atomic latin square of a composite order (namely 27) was previously known. We show that this one example can be generated by an established method of constructing latin squares using cyclotomic orthomorphisms in finite fields. The same method is used in this paper to construct atomic latin squares of composite orders 25, 49, 121, 125, 289, 361, 625, 841, 1369, 1849, 2809, 4489, 24649 and 39601. It is also used to construct many new atomic latin squares of prime order and perfect $1$-factorisations of the complete graph $K_{q+1}$ for many prime powers $q$. As a result, existence of such a factorisation is shown for the first time for $q$ in $\big\{$529, 2809, 4489, 6889, 11449, 11881, 15625, 22201, 24389, 24649, 26569, 29929, 32041, 38809, 44521, 50653, 51529, 52441, 63001, 72361, 76729, 78125, 79507, 103823, 148877, 161051, 205379, 226981, 300763, 357911, 371293, 493039, 571787$\big\}$. We show that latin squares built by the 'orthomorphism method' have large automorphism groups and we discuss conditions under which different orthomorphisms produce isomorphic latin squares. We also introduce an invariant called the train of a latin square, which proves to be useful for distinguishing non-isomorphic examples.

ON THE DENSITY OF DISCRIMINANTS OF ABELIAN EXTENSIONS OF A NUMBER FIELD

2010 β—½  
Vol 06 (06) β—½  
pp. 1273-1291
Author(s):  
BEHAILU MAMMO
Keyword(s):  
Asymptotic Formula β—½  
Galois Group β—½  
Number Field β—½  
Algebraic Number β—½  
Prime Order β—½  
Algebraic Number Field β—½  
Abelian Extensions β—½  
Cyclic Groups

Let G = Cβ„“ Γ— Cβ„“ denote the product of two cyclic groups of prime order β„“, and let k be an algebraic number field. Let N(k, G, m) denote the number of abelian extensions K of k with Galois group G(K/k) isomorphic to G, and the relative discriminant π’Ÿ(K/k) of norm equal to m. In this paper, we derive an asymptotic formula for βˆ‘m≀XN(k, G; m). This extends the result previously obtained by Datskovsky and Mammo.

scholarly journals HIGHER INDICATORS FOR SOME GROUPS AND THEIR DOUBLES

2012 β—½  
Vol 11 (02) β—½  
pp. 1250030 β—½  
Author(s):  
MARC KEILBERG
Keyword(s):  
Semidirect Product β—½  
Prime Order β—½  
Abelian Groups β—½  
Dihedral Groups β—½  
Cyclic Groups β—½  
Generalized Quaternion β—½  
Drinfel'd 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.

On (k, l)-sets in cyclic groups of odd prime order

2001 β—½  
Vol 63 (1) β—½  
pp. 115-121 β—½  
Author(s):  
T. Bier β—½  
A. Y. M. Chin
Keyword(s):  
Abelian Group β—½  
Cyclic Group β—½  
Arithmetic Progression β—½  
Prime Order β—½  
Finite Abelian Group β—½  
Maximum Cardinality β—½  
Cyclic Groups β—½  
Positive Integers

Let A be a finite Abelian group written additively. For two positive integers k, l with k β‰  l, we say that a subset S βŠ‚ A is of type (k, l) or is a (k, l) -set if the equation x1 + x2 + … + xk βˆ’ xk+1βˆ’β€¦ βˆ’ xk+1 = 0 has no solution in the set S. In this paper we determine the largest possible cardinality of a (k, l)-set of the cyclic group β„€P where p is an odd prime. We also determine the number of (k, l)-sets of β„€p which are in arithmetic progression and have maximum cardinality.

scholarly journals The Noether numbers for cyclic groups of prime order

2006 β—½  
Vol 207 (1) β—½  
pp. 149-155 β—½  
Author(s):  
P. Fleischmann β—½  
M. Sezer β—½  
R.J. Shank β—½  
C.F. Woodcock
Keyword(s):  
Prime Order β—½  
Cyclic Groups

EXTREMAL CAYLEY GRAPHS OF FINITE CYCLIC GROUPS

2008 β—½  
Vol 09 (01n02) β—½  
pp. 73-82
Author(s):  
JOSEPH J. LEE β—½  
ELYSIA J. SHEU β—½  
XINGDE JIA
Keyword(s):  
Finite Group β—½  
Cayley Graph β—½  
Nonempty Subset β—½  
Undirected Graph β—½  
Average Distance β—½  
Additive Group β—½  
Cyclic Groups β—½  
Residue Classes β—½  
Vertex Set β—½  
A Finite Group

Let Ξ“ be a finite group with a nonempty subset A. The Cayley graph Cay (Ξ“, A) of Ξ“ generated by A is defined as the digraph with vertex set Ξ“ and edge set {(x,y) | x-1 y ∈ A}. Cay (Ξ“, A) can be regarded as an undirected graph if x-1 ∈ A for all x ∈ A. Let [Formula: see text] denote the largest integer M so that there exists a set of integers A = {Β±1, Β±a2;…, Β±ak} such that the average distance between all pairs of vertices of Cay (β„€M,A) is at most r, where β„€M is the additive group of residue classes modulo M. It is proved in this paper that [Formula: see text] It is also proved that [Formula: see text]

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