Factoring groups into subsets

268 Pages·2009·1.28 MB·English

by  Sandor Szabo

Factoring Groups
into Subsets
Sándor Szabó
University of Pecs
Pecs, Hungary
Arthur D. Sands
University of Dundee
Dundee, United Kingdom
© 2009 by Taylor & Francis Group, LLC
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Contents
Symbol Descriptions ix
List of Tables xi
List of Figures xiii
Preface xv
1 Introduction 1
2 New factorizations from old ones 11
2.1 Restriction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3 Homomorphism . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.4 Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3 Non-periodic factorizations 37
3.1 Bad factorizations . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2 Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.3 Replacement . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4 Periodic factorizations 63
4.1 Good factorizations . . . . . . . . . . . . . . . . . . . . . . . 63
4.2 Good groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.3 Krasner factorizations . . . . . . . . . . . . . . . . . . . . . . 87
5 Various factorizations 93
5.1 The R´edei property . . . . . . . . . . . . . . . . . . . . . . . 93
5.2 Quasi-periodicity . . . . . . . . . . . . . . . . . . . . . . . . . 105
6 Factoring by many factors 121
6.1 Factoring periodic subsets . . . . . . . . . . . . . . . . . . . 121
6.2 Simulated subsets . . . . . . . . . . . . . . . . . . . . . . . . 128
7 Group of integers 141
7.1 Sum sets of integers . . . . . . . . . . . . . . . . . . . . . . . 141
7.2 Direct factor subsets . . . . . . . . . . . . . . . . . . . . . . . 146
7.3 Tiling the integers . . . . . . . . . . . . . . . . . . . . . . . . 152
vii
© 2009 by Taylor & Francis Group, LLC
viii Contents
8 Infinite groups 161
8.1 Groups with cyclic subgroups . . . . . . . . . . . . . . . . . . 161
8.2 Groups with special p-components . . . . . . . . . . . . . . . 169
9 Combinatorics 183
9.1 Complete maps . . . . . . . . . . . . . . . . . . . . . . . . . 183
9.2 Ramsey numbers . . . . . . . . . . . . . . . . . . . . . . . . . 189
9.3 Near factorizations . . . . . . . . . . . . . . . . . . . . . . . . 193
9.4 A family of random graphs . . . . . . . . . . . . . . . . . . . 199
9.5 Complex Hadamard matrices . . . . . . . . . . . . . . . . . . 201
10 Codes 207
10.1 Variable length codes . . . . . . . . . . . . . . . . . . . . . . 207
10.2 Error correcting codes . . . . . . . . . . . . . . . . . . . . . . 213
10.3 Tilings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
10.4 Integer codes . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
11 Some classical problems 235
11.1 Fuchs’s problems . . . . . . . . . . . . . . . . . . . . . . . . . 235
11.2 Full-rank factorizations . . . . . . . . . . . . . . . . . . . . . 239
11.3 Z-subsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
References 253
© 2009 by Taylor & Francis Group, LLC
Symbol Descriptions
Z The ring of integers. χ Characterofafiniteabelian
Z(n) The additive subgroup of group.
the residue classes modulo Kerχ The kernel of the character
n. χ.
Z(p∞) The Pru¨fer group, where p F (x) Thenthcyclotomicpolyno-
n
is a prime. mial.
G The order of the group G. degF The degree of the polyno-
| |
(G)p The p-component of the mial F.
group G, where p is a
R (r) A Ramsey number.
m
prime.
d(u,v) TheHammingdistanceofu
g The order of the element g
| | and v.
in a group.
S (u) An error sphere of radius e
e
(g) The H-component of g,
H centered at u.
where H is a subgroup.
ν(n) The number of the prime
(g) The p-component of g,
p factors of n.
where p is a prime.
[g,n] Bracket notation for a
GF(q) The Galois field of order q.
cyclic subset.
A The span of the set A in a
h i χ(Γ) The chromatic number of
group.
the graph Γ.
A The number of elements of
| | ω(Γ) The clique number of the
the subset A.
graph Γ.
Ann(A) The annihilator of the sub-
set A. A+B The sum of the subsets A
(A) The H-component of the and B.
H
subset A. A ϕB The composition of the
◦
(A) The p-component of the subsets A and B.
p
subset A, where p is a
prime.
ix
© 2009 by Taylor & Francis Group, LLC
List of Tables
1.1 Some difference sets . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 The lines of a finite projective plane . . . . . . . . . . . . . . 4
2.1 The factors D and E . . . . . . . . . . . . . . . . . . . . . . . 34
5.1 The factors A, H, and B . . . . . . . . . . . . . . . . . . . . . 97
5.2 The factors A, H, and B . . . . . . . . . . . . . . . . . . . . . 99
5.3 The factors A and B . . . . . . . . . . . . . . . . . . . . . . . 119
9.1 Complete maps . . . . . . . . . . . . . . . . . . . . . . . . . . 185
9.2 The incidence matrix of K . . . . . . . . . . . . . . . . . . . 191
13
9.3 The incidence matrix of the Keller graph K . . . . . . . . . 192
2
9.4 The values of s(i),t(i) . . . . . . . . . . . . . . . . . . . . . . 204
10.1 The factors A and B . . . . . . . . . . . . . . . . . . . . . . 211
1 1
10.2 Shor’s code . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
10.3 The divisors of 84. . . . . . . . . . . . . . . . . . . . . . . . . 221
10.4 A substitution error sphere . . . . . . . . . . . . . . . . . . . 227
10.5 A single shift error sphere . . . . . . . . . . . . . . . . . . . . 229
11.1 Equation relations among i, j, f(i), and f(j) . . . . . . . . . 241
xi
© 2009 by Taylor & Francis Group, LLC
List of Figures
1.1 A difference system in Z(21). . . . . . . . . . . . . . . . . . . 3
1.2 A closed polygon in the n=8 case. . . . . . . . . . . . . . . . 3
2.1 The implications. . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2 The subgroup H. . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.1 Cosets modulo H. . . . . . . . . . . . . . . . . . . . . . . . . 40
4.1 The vector rhombus. . . . . . . . . . . . . . . . . . . . . . . . 67
5.1 The cube tiling . . . . . . . . . . . . . . . . . . . . . . . . . . 104
9.1 The edges of K colored by A . . . . . . . . . . . . . . . . . 191
13 2
9.2 A Cayley graph. . . . . . . . . . . . . . . . . . . . . . . . . . 196
9.3 The Keller graph K .. . . . . . . . . . . . . . . . . . . . . . . 199
2
10.1 A (3,1)-semicross. . . . . . . . . . . . . . . . . . . . . . . . . 217
10.2 A (3,1)-cross. . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
11.1 Twins and not twins. . . . . . . . . . . . . . . . . . . . . . . . 237
11.2 A 2-dimensional cube tiling. . . . . . . . . . . . . . . . . . . . 238
xiii
© 2009 by Taylor & Francis Group, LLC
Preface
Thefactorizationtheoryofabeliangroupsdealswithdecomposinganabelian
group into a direct sum of its subsets. The nature of these problems are
partly algebraic and partly combinatorial. The results have applications in
various fields: geometry of tilings, variable length codes, integer codes, error
correctingcodes,cryptography,graphtheory,numbertheory,Fourieranalysis,
complex Hadamard matrices, just to mention a few.
This book is about the factorization theory of abelian groups focusing
mainlyoncyclicgroups. Thebook[157]treatsselectedtopicsofthefactoriza-
tion theory with anintentionto possibly touchallthe main areas. Inevitably
overlaps are bound to occur.
The direct sum of isomorphic copies of a cyclic group of prime order, that
is, an elementary p-group, behaves in many respects as a finite linear space
andmustbetreateddifferentlythanacyclicgroup. Someoftheresultsofthe
factorizationtheorybecometrivialwhenrestrictedtocyclicgroups. Onthese
occasions the exiting part of the problem lies in the non-cyclic case, while at
other times the substantial part of the result is in the cyclic case. It is an
empirical fact that factoring cyclic groups is still a rich field. For instance,
[85] claims that each cryptosystem in use is based on factoring a large cyclic
group. Manyofthe results andapplicationsareavailablewith the masteryof
simpler techniques than those required by the general non-cyclic case.
Withmercilessediting(notincluding manytopicsthathappenedto be our
personalfavorite)wekeptthematerialtoamanageablesize. Ourhopeisthat
this book will serve as a gentle introduction for practitioners of other fields
interested in the applications of the factorization theory of abelian groups.
xv
© 2009 by Taylor & Francis Group, LLC
Chapter 1
Introduction
Therearemanyproblemswithacombinatorialflavourrelatedtofiniteabelian
groups. ThefirstsuchresultismostlikelyduetoA.L.Cauchy[16]from1813
andrediscoveredby H. Davenport[25] in1935. Letp be a prime number and
let Z(p) be the additive group of residue classes modulo p. For two subsets
A, B of Z(p) we define A+B to be a+b : a A,b B and we use A
{ ∈ ∈ } | |
to denote the number of elements ofA. The Cauchy-Davenporttheoremnow
reads as follows.
If A, B are not empty subsets of Z(p), then
A+B A + B 1,
| |≥| | | |−
provided p A + B 1.
≥| | | |−
This result has since been extended and generalized in a number of ways.
(See the monograph [91].)
Let A be a set of Z(n) with A =s. The difference set
| |
D =A A= a a′ :a,a′ Z(n)
− { − ∈ }
is clearly symmetric in the sense that d D implies d D. There are at
∈ − ∈
most s(s 1)/2 distinct non-zero elements in D. The situation when each
−
non-zero element of Z(n) can be represented uniquely in the way
d=a a′, a,a′ A
− ∈
is particularly interesting. In this case one can use the elements of Z(n) as
points of a finite projective plane. The subsets A+b, b Z(n) will form
∈
the straight lines of the plane. In his book [62] F. K´arteszi considered a
regular polygon of n sides and coordinatized the vertices by the elements
0,1,...,n 1 of Z(n) walking around the polygon, say clockwise. (See also
−
the text [66].) Considering the distances between the vertex 0 and the other
vertices1,...,n 1makesclearthatthenumberofnon-zerodistinctdistances
−
occurring between the vertices of the polygon is (n 1)/2 when n is odd.
−
Therefore
s(s 1) n 1
− − .
2 ≤ 2
Wearelookingforthecaseswhenequationholds,thatis,whenn=s(s 1)+1.
−
Table 1.1 (on page 2) lists some values of n and s together with a suitable
choice of A where the bound is tight.
1
© 2009 by Taylor & Francis Group, LLC