Expo. Math. 24 (2006) 337 – 369
www.elsevier.de/exmath
Zero-sum problems in finite abelian groups: A survey
Weidong Gaoa , Alfred Geroldingerb,∗
a Center for Combinatorics, Nankai University, Tianjin 300071, PR China
b Institut für Mathematik und Wissenschaftliches Rechnen, Karl-Franzens Universität,
Heinrichstrasse 36, 8010 Graz, Austria
Received 19 December 2005
Abstract
We give an overview of zero-sum theory in finite abelian groups, a subfield of additive group theory
and combinatorial number theory. In doing so we concentrate on the algebraic part of the theory and
on the development since the appearance of the survey article by Y. Caro in 1996.
䉷 2006 Elsevier GmbH. All rights reserved.
MSC 2000: 11B50; 11P70; 11B75
Keywords: Zero-sum sequences; Finite abelian groups
1. Introduction
Let G be an additive finite abelian group. In combinatorial number theory a finite sequence
S = (g1 , . . . , gl ) = g1 · . . . · gl of elements of G, where the repetition of elements is
allowed and their order is disregarded, is simply called a sequence over G, and S is called
a zero-sum sequence if g1 + · · · + gl = 0. A typical direct zero-sum problem studies
conditions which ensure that given sequences have non-empty zero-sum subsequences
with prescribed properties. The associated inverse zero-sum problem studies the structure
of extremal sequences which have no such zero-sum subsequences.
∗ Corresponding author.
E-mail address: alfred.geroldinger@uni-graz.at (A. Geroldinger).
0723-0869/$ - see front matter 䉷 2006 Elsevier GmbH. All rights reserved.
doi:10.1016/j.exmath.2006.07.002
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W. Gao, A. Geroldinger / Expo. Math. 24 (2006) 337 – 369
These investigations were initiated by a result of P. Erdős, A. Ginzburg and A. Ziv, who
proved that 2n − 1 is the smallest integer l ∈ N such that every sequence S over a cyclic
group of order n has a zero-sum subsequence of length n (see [47]). Some years later,
P.C. Baayen, P. Erdős and H. Davenport (see [138,45,143]) posed the problem to determine
the smallest integer l ∈ N such that every sequence S over G of length |S| l has a zero-sum
subsequence. In subsequent literature that integer l has been called the Davenport constant
of G. It is denoted by D(G), and its precise value – in terms of the group invariants of G –
is still unknown in general.
These problems were the starting points for much research, as it turned out that questions of this type occur naturally in various branches of combinatorics, number theory
and geometry. Conversely, zero-sum problems have greatly influenced the development
of various subfields of these areas (among others, zero-sum Ramsey theory was initiated
by the works of A. Bialostocki and P. Dierker). So there are intrinsic connections with
graph theory, Ramsey theory and geometry (see [119,4,12,13] for some classical papers
and [11,10,105,14,108,40,123] for some recent papers). The following observation goes
back to H. Davenport: If R is the ring of integers of some algebraic number field with ideal
class group (isomorphic to) G, then D(G) is the maximal number of prime ideals (counted
with or without multiplicity) which occur in the prime ideal decomposition of aR for irreducible elements a ∈ R. Indeed, in the theory of non-unique factorizations it has turned
out that the monoid of all zero-sum sequences over G closely reflects the arithmetic of a
Krull monoid which has class group G and every class contains a prime (see [96, Corollary
3.4.12]). On the other hand, it was factorization theory which promoted the investigation of
inverse zero-sum problems, which appear naturally in that area. Apart from all that, zerosum problems occur in various types of number theoretical topics (as Carmicheal numbers
[1], Artin’s conjecture on additive forms [19] or permutation matrices [135]).
Zero-sum problems are tackled with a huge variety of methods. First of all we mention methods from additive group theory including all types of addition theorems (see
[96,136,137,142,112,107,109,103,133,124,9]). Furthermore, group algebras [74], results
from the covering area [164,132,80], from linear algebra [32,31] and polynomial methods
[2,3] play crucial roles. Moreover, in the meantime zero-sum theory has already developed
its own methods and a wealth of results which promote its further development.
The first survey article on zero-sum theory, written by Y. Caro, appeared 10 years ago
in 1996 (see [23,24]). The aim of the present article is to sketch the development in the
last decade and to give an overview over the present state of the area under the following
two restrictions. First, we do not outline the relationships to other areas, as graph theory, Ramsey theory or the theory of non-unique factorizations, but we restrict to what
is sometimes called the algebraic part of zero-sum theory. Second, although since the
1960s zero-sum problems were studied also in the setting of non-abelian groups (see
[36,146,150,147,148,170,63,39,172]), we restrict to the case of abelian groups. Since
Y. Caro’s article has an extended bibliography on the literature until 1994, we also refer to his bibliography and concentrate ourselves on papers having appeared since that
time. In Section 2, we fix our notations and terminology, and we give the definitions of the
key invariants. Then in the subsequent sections we present the state of knowledge on these
invariants and on the associated inverse problems.
Throughout this article, let G be an additive finite abelian group and let G• = G\{0}.
W. Gao, A. Geroldinger / Expo. Math. 24 (2006) 337 – 369
339
2. Preliminaries
Let N denote the set of positive integers, P ⊂ N the set of all prime numbers and let
N0 = N ∪ {0}. For integers a, b ∈ Z we set [a, b] = {x ∈ Z | a x b}, and for c ∈ N let
N c = N\[1, c − 1]. For a real number x, we denote by x the largest integer that is less
than or equal to x, and by x the smallest integer that is greater than or equal to x.
Throughout, all abelian groups will be written additively. For n ∈ N, let Cn denote a
cyclic group with n elements, and let nG = {ng | g ∈ G}. By the Fundamental Theorem of
Finite Abelian Groups we have
G
Cn1 · · · Cnr Cq1 · · · Cqs ,
where r = r(G) ∈ N0 is the rank of G, s = r∗ (G) ∈ N0 is the total rank of G,
n1 , . . . , nr ∈ N are integers with 1 < n1 | . . . | nr and q1 , . . . , qs are prime powers.
Moreover, n1 , . . . , nr , q1 , . . . , qs are uniquely determined by G, and we set
d∗ (G) =
r
i=1
(ni − 1)
and
k∗ (G) =
s
qi − 1
.
qi
i=1
Clearly, nr = exp(G) is the exponent of G, and if |G| = 1, then r(G) = d∗ (G) = k∗ (G) = 0
and exp(G) = 1.
Let s ∈ N. An s-tuple (e1 , . . . , es ) of elements of G is said to be independent if ei = 0
for all i ∈ [1, s] and, for every s-tuple (m1 , . . . , ms ) ∈ Zs ,
s
mi ei = 0 implies m1 e1 = · · · = ms es = 0.
i=1
An s-tuple (e1 , . . . , es ) of elements of G is called a basis if it is independent and G =
e1 · · · es .
We write sequences multiplicatively and consider them as elements of the free abelian
monoid over G, a point of view which was put forward by the requirements of the theory
of non-unique factorizations. Thus, we have at our disposal all notions from elementary
divisibility theory which provides a suitable framework when dealing with subsequences
of given sequences, and we may apply algebraic concepts in a natural way.
Let F(G) be the free abelian monoid, multiplicatively written, with basis G. The elements
of F(G) are called sequences over G. We write sequences S ∈ F(G) in the form
S=
g vg (S) with vg (S) ∈ N0 for all g ∈ G.
g∈G
We call vg (S) the multiplicity of g in S, and we say that S contains g, if vg (S) > 0. S is
called squarefree if vg (S)1 for all g ∈ G. The unit element 1 ∈ F(G) is called the
empty sequence. A sequence S1 is called a subsequence of S if S1 | S in F(G) (equivalently,
vg (S1 )vg (S) for all g ∈ G), and it is called a proper subsequence of S if it is a subsequence
with 1 = S1 = S. If a sequence S ∈ F(G) is written in the form S = g1 · . . . · gl , we tacitly
assume that l ∈ N0 and g1 , . . . , gl ∈ G.
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W. Gao, A. Geroldinger / Expo. Math. 24 (2006) 337 – 369
For a sequence
S = g1 · . . . · g l =
g vg (S) ∈ F(G),
g∈G
we call
|S| = l =
vg (S) ∈ N0
the length of S,
g∈G
h(S) = max{vg (S) | g ∈ G} ∈ [0, |S|] the maximum of the multiplicities of S,
k(S) =
l
1
∈ Q0
ord(gi )
i=1
the cross-number of S,
supp(S) = {g ∈ G | vg (S) > 0} ⊂ G
(S) =
l
gi =
i=1
k (S) =
vg (S)g ∈ G
the support of S,
the sum of S,
g∈G
i∈I
gi I ⊂ [1, l] with |I | = k
the set of k-term subsums of S, for all k ∈ N,
k (S) =
j (S), k (S) =
j (S),
j ∈[1,k]
j k
and
(S) = 1 (S) the set of (all) subsums of S.
The sequence S is called
• zero-sumfree if 0 ∈
/ (S),
• a zero-sum sequence if (S) = 0,
• a minimal zero-sum sequence if it is a non-empty zero-sum sequence and every proper
subsequence is zero-sumfree,
• a short zero-sum sequence if it is a zero-sum sequence of length |S| ∈ [1, exp(G)].
We denote by B(G) the set of all zero-sum sequences and by A(G) the set of all minimal
zero-sum sequences. Then B(G) ⊂ F(G) is a submonoid (also called the block monoid
over G); it is a Krull monoid and A(G) is the set of atoms of B(G) (see [96, Proposition
2.5.6]). For any map of abelian groups : G → G , there exists a unique homomorphism
: F(G) → F(G ) with | G = . Usually, we simply write instead of . Explicitly,
: F(G) → F(G ) is given by (g1 · . . . · gl ) = (g1 ) · . . . · (gl ) for all l ∈ N0
and g1 , . . . , gl ∈ G. If S ∈ F(G), then |(S)| = |S| and supp((S)) = (supp(S)). If
: G → G is even a homomorphism, then ((S)) = ((S)), ((S)) = ((S)) and
W. Gao, A. Geroldinger / Expo. Math. 24 (2006) 337 – 369
341
(B(G)) ⊂ B(G ). In particular, we use the inversion (g → −g) and the translation
(g → g0 + g), and for S = g1 · . . . · gl ∈ F(G) we set
−S = (−g1 ) · . . . · (−gl )
and
g0 + S = (g0 + g1 ) · . . . · (g0 + gl ) ∈ F(G).
If g ∈ G is a non-zero element and
S = (n1 g) · . . . · (nl g),
where l ∈ N0 and n1 , . . . , nl ∈ [1, ord(g)],
then
Sg =
n 1 + · · · + nl
ord(g)
is called the g-norm of S. If S is a zero-sum sequence for which {0} = supp(S) ⊂ G is
a finite cyclic group, then
ind(S) = min{Sg | g ∈ G
with supp(S) = g} ∈ N0
is called the index of S. We set ind(1) = 0, and if supp(S) = {0}, then we set ind(S) = 1.
Next we give the definition of the zero-sum invariants which we are going to discuss in
the subsequent sections. We concentrate on invariants dealing with general sequences, as
introduced in Definition 2.1. However, by an often used technique, problems on general
sequences are reduced to problems on squarefree sequences, and thus we briefly deal also
with invariants on squarefree sequences (or in other words, with sets), as introduced in
Definition 2.2.
Definition 2.1. Let exp(G) = n and k, m ∈ N with k exp(G). We denote by
• D(G) the smallest integer l ∈ N such that every sequence S ∈ F(G) of length |S| l
has a non-empty zero-sum subsequence. The invariant D(G) is called the Davenport
constant of G.
• d(G) the maximal length of a zero-sumfree sequence over G.
• (G) the smallest integer l ∈ N such that every sequence S ∈ F(G) of length |S| l
has a short zero-sum subsequence.
• smn (G) the smallest integer l ∈ N such that every sequence S ∈ F(G) of length |S| l
has a zero-sum subsequence T of length |T | = mn. In particular, we set s(G) = sn (G).
• snN (G) the smallest integer l ∈ N such that every sequence S ∈ F(G) of length |S| l
has a non-empty zero-sum subsequence T of length |T | ≡ 0 mod n.
• Ek (G) the smallest integer l ∈ N such that every sequence S ∈ F(G) of length |S| l
has a zero-sum subsequence T with k|T |.
• (G) the smallest integer l ∈ N0 with the following property:
For every zero-sumfree sequence S ∈ F(G) of length |S| l there exist a subgroup
H ⊂ G and an element a ∈ G\H such that G• \(S) ⊂ a + H .
A simple argument (see [96, Section 5.1] for details) shows that
d(G) = max{|S| | S ∈ F(G), (S) = G• } and
1 + d(G) = D(G) = max{|S||S ∈ A(G)}.
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W. Gao, A. Geroldinger / Expo. Math. 24 (2006) 337 – 369
Definition 2.2. We denote by
• Ol(G) the smallest integer l ∈ N such that every squarefree sequence S ∈ F(G) of
length |S| l has a non-empty zero-sum subsequence. The invariant Ol(G) is called the
Olson constant of G.
• ol(G) the maximal length of a squarefree zero-sumfree sequence S ∈ F(G).
• cr(G) the smallest integer l ∈ N such that every squarefree sequence S ∈ F(G• ) of
length |S|l satisfies (S) = G. The invariant cr(G) is called the critical number of G.
• g(G) the smallest integer l ∈ N such that every squarefree sequence S ∈ F(G) of length
|S| l has a zero-sum subsequence T of length |T | = exp(G).
We use the convention that min(∅) = sup(∅) = 0. For a subset G0 ⊂ G and some integer
l ∈ N, R.B. Eggleton and P. Erdős (see [41]) introduced the f-invariant
f(G0 , l) = min{|(S)| | S ∈ F(G0 ), S squarefree and zero-sumfree, |S| = l}.
The basic relationships between these invariants are summarized in Lemma 10.1.
3. On the Davenport constant
Let G = Cn1 · · · Cnr with 1 < n1 | . . . | nr , r = r(G) and let (e1 , . . . , er ) be a basis
of G with ord(ei ) = ni for all i ∈ [1, r]. Then the sequence
S=
r
eini −1 ∈ F(G)
i=1
is zero-sumfree whence we have the crucial inequality
d(G)d∗ (G).
In the 1960s, D. Kruyswijk and J.E. Olson proved independently the following result (see
[143,5,44,144] and [96, Theorems 5.5.9 and 5.8.3]).
Theorem 3.1. If G is a p-group or r(G) 2, then d(G) = d∗ (G).
We present two types of results implying that d(G) = d∗ (G). The first one is due to
P. van Emde Boas et al. (see [44, Theorems 3.9, 4.2], where more results of this flavor
may be found) and the second is due to S.T. Chapman et al. (see [25], and also the various
conjectures in that paper).
Theorem 3.2. Let G = C2n1 C2n2 C2n3 and H = Cn1 Cn2 Cn3 with 1 n1 | n2 | n3 .
If (H ) = d∗ (H ) − 1, then d(G) = d∗ (G).
Theorem 3.3. Let G=H Ckm where k, m ∈ N and H ⊂ G is a subgroup with exp(H )|m.
If d(H Cm ) = d(H ) + m − 1 and (H Cm ) d(H ) + 2m, then d(G) = d(H ) + km − 1.
W. Gao, A. Geroldinger / Expo. Math. 24 (2006) 337 – 369
343
In particular (use Theorem 3.1 and [96, Proposition 5.7.7]), if m is a prime power and
d(H ) < m, then d(G) = d∗ (G).
These and similar results give rise to long lists of explicit groups satisfying d(G) =
d∗ (G) (see [44,25,6,46,35]). The first example of a group G with d(G) > d∗ (G) is due to
P.C. Baayen. In [44, Theorem 8.1] it is shown that
d(G) > d∗ (G) for G = C24k C4k+2 with k ∈ N,
and more examples are given in [46]. Let H ⊂ G be a subgroup. Then d(H ) + d(G/H )
d(G), and if G is as above, I ⊂ [1, r] and
H=
Cni then d(H ) > d∗ (H ) implies d(G) > d∗ (G)
i∈I
(see [96, Proposition 5.1.11]). This shows that the interesting groups with d(G) > d∗ (G)
are those with small rank. A. Geroldinger and R. Schneider showed that there are infinitely
many G with r(G) = 4 such that d(G) > d∗ (G). The following result may be found in [98]
and [77, Theorem 3.3].
Theorem 3.4. We have d(G) > d∗ (G) in each of the following cases:
1. G = Cm Cn2 C2n , where m, n ∈ N 3 are odd and m | n.
5−i
, where n ∈ N 3 is odd and i ∈ [2, 4].
2. G = C2i C2n
Let G = C2r Cn where r ∈ N and n ∈ N 3 is odd. Then d(G) = d∗ (G) if and only if
r 4 (see [98, Corollary 2]). For some small r 5 and n 3 the precise value of d(G) was
recently determined in [49]. The growth of d(G) − d∗ (G) is studied in [140].
We make the following conjecture.
Conjecture 3.5. If r(G) = 3 or G = Cnr with n, r ∈ N 3 , then d(G) = d∗ (G).
For groups of rank three Conjecture 3.5 goes back to P. van Emde Boas (see [46]) and is
supported by [69]. For groups of the form G = Cnr it is supported by [80, Theorem 6.6].
The next result provides upper bounds on D(G). The first one is due to P. van Emde Boas
and D. Kruyswijk [46, Theorem 7.1] and is sharp for cyclic groups (for other approaches
and related bounds see [8,141]). The second bound is sharp for groups of rank 2 and with
H = pG for some prime divisor p of exp(G) (see [96, Theorem 5.5.5 and Proposition
5.7.11]).
Theorem 3.6. Let exp(G) = n 2 and H ⊂ G be a subgroup.
1. d(G)(n − 1) + n log |G|
n .
2. d(G)d(H ) exp(G/H ) + max{d(G/H ), (G/H ) − exp(G/H ) − 1}.
We end this section with a conjecture supported by [96, Theorem 6.2.8].
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W. Gao, A. Geroldinger / Expo. Math. 24 (2006) 337 – 369
Conjecture 3.7. If |G| > 1, then D(G) d∗ (G) + r(G).
4. On the structure of long zero-sumfree sequences
Let S ∈ F(G) be a zero-sumfree sequence of length |S| = d(G). According to general
philosophy in inverse additive number theory (see [142,53,54]), S should have some structure. Obviously, if G is cyclic of order n2, then S = g n−1 for some g ∈ supp(S) with
ord(g) = n, and if S is an elementary 2-group of rank r, then S = e1 · . . . · er for some basis
(e1 , . . . , er ) of G. Apart from these trivial cases very little is known up to now. The most
modest questions one could ask are the following:
1. What is the order of elements in supp(S)?
2. What is the multiplicity of elements in supp(S)? What is a reasonable lower bound for
h(S)?
3. How large is supp(S)?
Crucial in all investigations of zero-sumfree sequences is the following inequality of
Moser-Scherk (see [96, Theorem 5.3.1]): Let S ∈ F(G) be a zero-sumfree sequence.
If S = S1 S2
then |(S)| |(S1 )| + |(S2 )|.
By M. Freeze and W.W. Smith ([52, Theorem 2.5, 96, Proposition 5.3.5]) this implies that
|(S)|2|S| − h(S)|S| + |supp(S)| − 1.
We start with the following conjecture.
Conjecture 4.1. Every zero-sumfree sequence S ∈ F(G) of length |S| = d(G) has some
element g ∈ supp(S) with ord(g) = exp(G).
The conjecture is true for cyclic groups, p-groups (see [96, Corollary 5.1.13]), groups of
the form G = Cn Cn (see below) and for G = C2 C2n (see [78]). As concerns the second
question, the philosophy is that in groups where the exponent is large in comparison with
the rank, h(S) should be large.
For cyclic groups, there are the following results going back to J.D. Bovey, P. Erdős,
I. Niven, W. Gao, A. Geroldinger and Y. ould Hamidoune (see [17,76,97] and [96, Theorem
5.4.5]).
Theorem 4.2. Let G be cyclic of order n 3, and let S ∈ F(G) be a zero-sumfree sequence
of length
|S|
n+1
.
2
1. For all g ∈ supp(S) we have ord(g)3.
2. There exists some g ∈ supp(S) with vg (S)2|S| − n + 1.
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345
3. There exists some g ∈ supp(S) with ord(g) = n such that
vg (S)
n+5
6
if n is odd
and vg (S) 3 if n is even.
In cyclic groups long zero-sumfree sequences and long minimal zero-sum sequences can
be completely characterized (see [71]).
Theorem 4.3. Let G by cyclic of order n 2 and let S ∈ F(G) be a zero-sumfree sequence
of length |S| = n − k with k ∈ [1, n/3] + 1]. Then there exists some g ∈ G with ord(g) = n
and x1 , . . . , xk−1 ∈ [1, n − 1] such that
S=g
n−2k+1
k−1
k−1
i=1
i=1
(xi g) and
xi 2k − 2.
In particular, every minimal zero-sum sequence S ∈ A(G) of length |S| n − n/3 has
ind(S) = 1.
The index of zero-sum sequences over cyclic groups is investigated in [71,26,29].
In [127] (p. 344 with d = n) it is conjectured that every sequence S ∈ F(Cn ) of length
|S| = n has a non-empty zero-sum subsequence T with ind(T ) = 1. Among others, the
g-norm and the index of zero-sum sequences play a role in arithmetical investigations (see
[96, Section 6.8]).
Next we discuss groups of the form G=Cn Cn with n 2 (see [77,167,79], [96, Section
5.8] and [131]).
Theorem 4.4. Let G = Cn Cn with n2. Then the following statements are equivalent:
(a) If S ∈ F(G), |S| = 3n − 3 and S has no zero-sum subsequence T of length |T | n,
then there exists some a ∈ G such that 0n−1 a n−2 | S.
(b) If S ∈ F(G) is zero-sumfree and |S| = d(G), then a n−2 | S for some a ∈ G.
(c) If S ∈ A(G) and |S| = D(G), then a n−1 | S for some a ∈ G.
(d) If S ∈ A(G) and |S| = D(G), then there exists a basis (e1 , e2 ) of G and integers
x1 , . . . , xn ∈ [0, n − 1] with x1 + · · · + xn ≡ 1 mod n such that
S = e1n−1
n
(x e1 + e2 ).
=1
Moreover, if S ∈ A(G) and |S| = D(G), then ord(g) = n for every g ∈ supp(S), and if
n is prime, then |supp(S)| ∈ [3, n].
Conjecture 4.5. For every n2 the four equivalent statements of Theorem 4.4 are
satisfied.
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W. Gao, A. Geroldinger / Expo. Math. 24 (2006) 337 – 369
Conjecture 4.5 has been verified for n ∈ [2, 7], and if it holds for some n6, then it holds
for 2n (see [79, Theorem 8.1]). We continue with a result for non-cyclic groups having large
exponent (see [79]).
Theorem 4.6.
1. Let G = Cn1 Cn2 with 1 < n1 | n2 and n2 > n1 (n1 + 1). Let : G → G = Cn1 Cn1 be
the canonical epimorphism and S ∈ A(G) of length |S| = D(G). If g k |(S) for some
k > n1 and some g ∈ G, then g k |S for some g ∈ −1 (g).
2. Let G = H Cn where exp(G) = n = lm, H ⊂ G a subgroup with exp(H ) | m, m 2
and l 4|H | > 4(m − 2). Let : G → G = H Cm denote the canonical epimorphism
and S ∈ F(G) a zero-sumfree sequence of length |S| = n. Then S has a subsequence
T of length |T |(l − 2|H | + 1)m such that the following holds: If g k |(T ) for some
k > m and some g ∈ G, then g k |T for some g ∈ −1 (g).
For general finite abelian groups there is the following result (see [76], [96, Theorem
5.3.6]) which plays a key role in the proof of Theorem 10.4.2).
Theorem 4.7. Let G0 ⊂ G be a subset, k ∈ N and k 2 be such that f(G0 , k) > 0, and let
S ∈ F(G0 ) be a zero-sumfree sequence of length
|S| |G| − k
+ 1 k.
f(G0 , k)
Then there exists some g ∈ G0 such that
vg (S)
|G| − k − 1
|S|
−
.
k − 1 (k − 1)f(G0 , k)
If the rank of the group is large in comparison with the exponent, there is in general no element with high multiplicity (see Theorem 10.4.1), but in case of elementary
p-groups there is the following structural result (see [77, Theorem 10.3], [80, Corollary 6.3],
[96, Corollary 5.6.9]).
Theorem 4.8. Let G be a finite elementary p-group and S ∈ F(G) be a zero-sumfree
sequence of length |S| = d(G). Then (g, h) is independent for any two distinct elements
g, h ∈ supp(S).
We continue with the following
Conjecture 4.9. Let G = Cn1 · · · Cnr with 1 < n1 | · · · |nr , k ∈ [1, n1 − 1] and S ∈
F(G) be a sequence of length |S| = k + d(G). If S has no zero-sum subsequence S of
length |S | > k, then S = 0k T where T ∈ F(G) is zero-sumfree.
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347
The following example shows that in Conjecture 4.9 the restriction k ∈ [1, n1 − 1] is
essential:
Let T ∈ F(G) be a zero-sumfree sequence of length |T | = d(G) such that vg (S) =
ord(g) − 1 for some g ∈ G. Then for every l ∈ N the sequence S = g lord(g) T has no
zero-sum subsequence S of length |S | > lord(g).
Next we discuss the invariant (G) which was introduced by P. van Emde Boas in connection with his investigations of the Davenport constant for groups of rank three (see
[44, p. 15] and [69]). An easy argument (see [96, Proposition 5.1.16]) shows that
d(G) − 1(G) d(G)
and we make the following conjecture.
Conjecture 4.10. (G) = d(G) − 1.
The following result goes back to P. van Emde Boas, W. Gao and A. Geroldinger
([44, Theorem 2.8], [79, Theorem 5.3], [96, Theorems 5.5.9 and 5.8.10], for more see also
[69, Theorem 5.2]).
Theorem 4.11. Conjecture 4.10 holds true in each of the following cases:
1. G is cyclic.
2. G is a p-group.
3. G = Cn Cn satisfies Conjecture 4.5.
We end this section with a result (see [81]) showing that minimal zero-sum sequences
are not additively closed (apart from some well-defined exceptions).
Theorem 4.12. Let S ∈ F(G• ) be a sequence of length |S| 4, and let S = BC with
B, C ∈ F(G) such that |B||C|. If (T ) ∈ supp(S) for all subsequences T of B with
|T | = 2 and for all subsequences T of C with |T | = 2, then S has a proper zero-sum
subsequence, apart from the following exceptions:
1. |C| = 1, and we are in one of the following cases:
(a) B = g k and C = 2g for some k 3 and g ∈ G with ord(g) k + 2.
(b) B = g k (2g) and C = 3g for some k 2 and g ∈ G with ord(g) k + 5.
(c) B = g1 g2 (g1 + g2 ) and C = g1 + 2g2 for some g1 , g2 ∈ G with ord(g1 ) = 2 and
ord(g2 )5.
2. {B, C} = {g(9g)(10g), (11g)(3g)(14g)} for some g ∈ G with ord(g) = 16.
If S = g1 · . . . · gl ∈ F(G) such that ord(gk ) > k k for all k ∈ [1, l], then G. Harcos and
I. Ruzsa showed that S allows a product decomposition S = S1 S2 where S1 and S2 are both
zero-sumfree (see [120]).
5. On generalizations of the Davenport constant
We discuss two generalizations of the Davenport constant in some detail (for yet another generalization, the barycentric Davenport constant, we refer to [34]). The first one
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was introduced by F. Halter-Koch in connection with the analytic theory of non-unique
factorizations (see [110]).
Definition 5.1. Let k ∈ N. We denote by
• Dk (G) the smallest integer l ∈ N such that every sequence S ∈ F(G) of length |S| l
is divisible by a product of k non-empty zero-sum sequences.
• dk (G) the largest integer l ∈ N with the following property:
There is a sequence S ∈ F(G) of length |S| = l which is not divisible by a product
of k non-empty zero-sum sequences.
Obviously, we have Dk (G) = 1 + dk (G), d1 (G) = d(G) and D1 (G) = D(G). We present
one result on dk (G) which, among others, may be found in [96, Section 6.1].
Theorem 5.2. Let exp(G) = n and k ∈ N.
1. Let G = H Cn where H ⊂ G is a subgroup. Then
d(H ) + kn − 1 dk (G) (k − 1)n + max{d(G), (G) − n − 1}.
In particular, if d(G) = d(H ) + n − 1 and (G) d(G) + n + 1, then dk (G) = d(G) +
(k − 1)n.
2. If r(G)2, then dk (G) = d(G) + (k − 1)n.
3. If G a p-group and D(G) 2n − 1, then dk (G) = d(G) + (k − 1)n.
The following generalization of the Davenport constant was introduced by M. Skałba in
connection with his investigations on binary quadratic forms (see [160–162]).
Definition 5.3. For every g ∈ G, let Dg (G) denote the largest integer l ∈ N with the
following property:
There is a sequence S ∈ F(G) of length |S| = l and sum (S) = g such that every
proper subsequence of S is zero-sumfree.
By definition, D0 (G) = D(G), and if g = 0, then Dg (G)d(G). The following result is
due to M. Skałba (see [161, Theorem 2] and [162, Theorem 1])
Theorem 5.4. Let G=Cn1 Cn2 with 1 n1 | n2 and (e1 , e2 ) a basis of G. Let g = a1 e1 +
a2 e2 ∈ G• with a1 ∈ [0, n1 − 1], a2 ∈ [0, n2 − 1] and d = gcd(gcd(a1 , n1 ), gcd(a2 , n2 )).
Then
n 1 + n2 − d − 1
if d = n1 ,
Dg (G) =
n1 + n2 − gcd(a2 , n2 ) − 1 if d = n1 .
Lemma 5.5. Let exp(G) = n2. Then the following statements are equivalent:
(a) There exists some g ∈ G with ord(g) = n such that Dg (G) = d(G).
(b) For all g ∈ G with ord(g) = n we have Dg (G) = d(G).
(c) There exists a minimal zero-sum sequence S ∈ F(G) of length |S| = D(G) such that
max{ord(g) | g ∈ supp(S)} = n.
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Proof. (a) ⇒ (b) Let g, g ∗ ∈ G with ord(g)=ord(g ∗ )=n and suppose that Dg ∗ (G)=d(G).
Then there exists a zero-sumfree sequence S ∈ F(G) of length |S| = d(G) and (S) = g ∗ .
If : G → G is a group automorphism with (g ∗ )=g, then (S) is a zero-sumfree sequence
of length |(S)| = d(G) and ((S)) = ((S)) = g whence Dg (G) = d(G).
(b) ⇒ (c) Let g ∈ G and S ∈ F(G) a zero-sumfree sequence with (S) = g and
|S| = Dg (G) = d(G). Then the sequence S ∗ = (−g)S has the required properties.
(c) ⇒ (a) Assume to the contrary that for all g ∈ G with ord(g) = n we have
Dg (G) < d(G). This means that for all zero-sumfree sequences S ∈ F(G) with |S| = d(G)
we have ord((S)) < n. But this implies that for all minimal zero-sum sequences S ∈ F(G)
of length |S| = D(G) we have max{ord(g) | g ∈ supp(S)} < n, a contradiction. Note that Conjecture 4.1 implies Condition (c) of Lemma 5.5. Using this condition we
immediately obtain the following corollary.
Corollary 5.6. If d∗ (G) = d(G) and g ∈ G with ord(g) = exp(G), then Dg (G) = d(G).
6. On the invariants (G), s(G) and their analogues
We start with a key result first obtained by W. Gao (see [60]). Its proof is based on the
Addition Theorem of Kemperman–Scherk (for the version below we refer to [96, Theorem
5.7.3]).
Theorem 6.1. Let S ∈ F(G) be a sequence of length |S| |G|. Then S has a non-empty
zero-sum subsequence T of length |T | min{h(S), max{ord(g) | g ∈ supp(S)}}.
Now we discuss the invariants (G), s(G) and their relationship. Both invariants have
received a lot of attention in the literature. The various contributions and the present state
of knowledge are well-described in [40], where also the connection with finite geometry is
discussed (see also [82]). Therefore we only mention some of the most recent results, and
then we discuss the relationship of (G) and s(G) in greater detail. A simple observation
shows that
D(G)(G) s(G) − exp(G) + 1.
Using Theorem 6.1 we obtain the following upper bounds on (G) and s(G) (see [96,
Theorem 5.7.4]) which are sharp for cyclic groups.
Theorem 6.2. (G) |G| and s(G) |G| + exp(G) − 1.
Both invariants, (G) and s(G) are completely determined for groups of rank at most
two (see [96, Theorem 5.8.3]). Theorem 6.3 is based on the result by C. Reiher which states
that s(Cp Cp ) = 4p − 3 for all p ∈ P (see [155], and also [156]), and it contains the
Theorem of Erdős–Ginzburg–Ziv (set n1 = 1). Theorem 6.4 may be found in [159].
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Theorem 6.3. Let G = Cn1 Cn2 with 1 n1 | n2 . Then
(G) = 2n1 + n2 − 2
and
s(G) = 2n1 + 2n2 − 3.
Theorem 6.4. Let G be a p-group for some odd prime p with exp(G)=n and D(G)2n−1.
Then
2D(G) − 1 (G) + n − 1 s(G)D(G) + 2n − 2.
In particular, if D(G) = 2n − 1, then s(G) = (G) + n − 1 = 4n − 3.
We continue with the following conjecture
Conjecture 6.5. (G) = s(G) − exp(G) + 1.
Theorem 6.6. Conjecture 6.5 holds true in each of the following cases:
1.
2.
3.
4.
exp(G) ∈ {2, 3, 4}.
r(G)2.
G is a p-group for some odd prime p and D(G) = 2 exp(G) − 1.
G = C53 .
Proof. 1. Is proved in [73], 2. follows from Theorem 6.3, and 3. follows from Theorem 6.4.
In order to give an idea of the arguments we are going to prove 4. We need the following
two results:
F1. If n ∈ N 3 is odd, then (Cn3 ) 8n − 7 (this is due to C. Elsholtz [43], see also
[40, Lemma 3.4]).
F2. If exp(G) = n and S ∈ F(G) such that
|S|(G) + n − 1
and
h(S)n − n/2 − 1,
then S has a zero-sum subsequence of length n (see [73, Proposition 2.7]).
Let G = C53 . It suffices to show that s(G) (G) + 4. Let S ∈ F(G) be a sequence of
length (G) + 4. We have to verify that S has a zero-sum subsequence of length 5. By F1
we have, |S|37. If we can prove that h(S) 2, then the assertion follows from F2.
Assume to the contrary that S is squarefree. Let G = H g where H ⊂ G is a subgroup
with |H | = 25 and g ∈ G with ord(g) = 5. Then
S=
l
i=1
(gi + hi ),
where gi ∈ g, hi ∈ H, and we set T =
l
gi .
i=1
If h(T )9, say g1 = · · · = g9 , then h1 , . . . , h9 are pairwise distinct. Since g(C52 ) = 9 (see
[125] and Conjecture 10.2), the sequence h1 ·. . .·h9 has a zero-sum subsequence of length 5,
and therefore S has a zero-sum subsequence of length 5.
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351
Suppose that h(T )8. Then T = 0l0 g l1 (2g)l2 (3g)l3 (4g)l4 with l0 , l1 , l2 , l3 , l4 ∈ [5, 8],
and we write S in the form
S=
li
4 (ig + hi,j ) with all hi,j ∈ H .
i=0 j =1
Since S is squarefree, for every i ∈ [0, 4] the elements hi,1 , . . . , hi,li are pairwise distinct,
and we set Ai = {hi,1 , . . . , hi,li }. Note that 0 + g + 2g + 3g + 4g = 0 ∈ G. So if
0 ∈ A = A 0 + A1 + A2 + A3 + A4 ,
then S has a zero-sum subsequence of length 5. Let K be the maximal subgroup of H such
that A + K = A. By Kneser’s Addition Theorem (see [96, Theorem 5.2.6.2]) we obtain that
|A| 4
|Ai + K| − 4|K|.
i=0
If |K| = 1, then |A| |A0 | + |A1 | + |A2 | + |A3 | + |A4 | − 4 = |S| − 4 33, a contradiction.
Assume to the contrary that |K| = 5. Since |A0 | + |A1 | + |A2 | + |A3 | + |A4 | = |S| 37
and |Ai | = li ∈ [5, 8], it follows that |Ai |6 for at least four indices i ∈ [0, 4]. Therefore
we obtain that
|A|
4
|Ai + K| − 4|K|4 · 2|K| + |K| − 4|K| = 5|K| = 25,
i=0
a contradiction. Thus it follows that K = H whence A = H and we are done.
For recent progress on Conjecture 6.5 we refer to [82]. Next we consider the invariant
snN (G). Theorem 6.3 allows to determine snN (G) for groups G of rank r(G)2.
Theorem 6.7. Let exp(G) = n 2.
1. d(G) + n snN (G) min{s(G), D(GCn )}.
2. We have snN (G) = d(G) + n in each of the following cases:
(a) G is a p-group.
(b) G = Cn1 Cn2 with 1 n1 | n2 .
Proof. 1. is simple (see [79, Lemma 3.5]) and 2.(a) is a consequence of 1. To verify 2.(b),
let G = Cn1 Cn2 with 1n1 | n2 . Then 1. implies that d(G) + n2 snN (G) whence it
remains to prove that snN (G) d(G) + n2 = n1 + 2n2 − 2. If n1 = 1, this follows from 1.
Let S ∈ F(G) be a sequence of length |S| = n1 + 2n2 − 2. We have to show that S has a
zero-sum subsequence of length n2 or 2n2 .
Let H = GCn2 = G
e with ord(e) = n2 , so that every h ∈ GCn2 has a unique
representation h = g + j e, where g ∈ G and j ∈ [0, n2 − 1]. We define : G → H by
(g) = g + e for every g ∈ G. Thus it suffices to show that (S) has a non-empty zero-sum
subsequence. We distinguish two cases.
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Case 1: n1 = n2 .
We set n = n1 and proceed by induction on n. If n is prime, the assertion follows from
2.(a). Suppose that n is composite, p a prime divisor of n and : H → H the multiplication
by p. Then pG
Cn/p Cn/p and Ker()
Cp3 . Since s(pG) = 4(n/p) − 3 and |S| = 3n −
2 (3p − 4)(n/p) + 4n/p − 3, S admits a product decomposition S = S1 · . . . · S3p−3 S such that, for all i ∈ [1, 3p − 3], (Si ) has sum zero and length |Si | = n/p (for details see
[96, Lemma 5.7.10]). Then |S | = 3n/p − 2 = snN (Cn/p Cn/p ), and thus S has a subsequence S3p−2 such that (S3p−2 ) has sum zero and length |S3p−2 | ∈ {n/p, 2n/p}. This
implies that
3p−2
((Si )) ∈ F(Ker()).
i=1
Since D(Ker()) = 3p − 2, there exists a non-empty subset I ⊂ [1, 3p − 2] such that
(Si )
((Si )) = 0 whence
i∈I
i∈I
is a non-empty zero-sum subsequence of (S).
Case 2: n2 > n1 .
2
2
Let m=n−1
1 n2 and let : H =Cn1 Cn2 → Cn1 mC n2 be a map which is the identity on
the first component and the multiplication by m on the second and on the third component
whence Ker()
Cm Cm and (G)
Cn1 Cn1 . Since s(Cn1 Cn1 ) = 4n1 − 3 and |S| =
n1 +2n2 −2 (2m−3)n1 +(4n1 −3), S admits a product decomposition S=S1 ·. . .·S2m−2 S ,
where for all i ∈ [1, 2m − 2], (Si ) has sum zero and length |Si | = n1 . Then |S | = 3n1 − 2,
and since by Case 1, snN (Cn1 Cn1 ) = 3n1 − 2, the sequence S has a subsequence S2m−1
such that (S2m−1 ) has sum zero and length |S2m−1 | ∈ {n1 , 2n1 }. This implies that
2m−1
((Si )) ∈ F(Ker()).
i=1
Since D(Ker()) = 2m − 1, there exists a non-empty subset I ⊂ [1, 2m − 1] such that
((Si )) = 0 whence
(Si )
i∈I
i∈I
is a non-empty zero-sum subsequence of (S).
Next we deal with zero-sum subsequences of length |G|. The following result is due
to W. Gao and Y. Caro (see [21,22,62] and also [96, Proposition 5.7.9]). In Section 9, we
discuss generalizations due to Y. ould Hamidoune. The structure of sequences S of length
|S| = |G| + d(G) − 1 which have no zero-sum subsequence of length |G| is studied in [94].
Theorem 6.8. s|G| (G) = |G| + d(G).
Note that Theorem 6.8 yields immediately a generalization of a Theorem of Hall (see
[135, Section 3]).
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353
Conjecture 6.9. Let G be cyclic of order n2, q the smallest primedivisor of n and
S ∈ F(G• ) be a sequence of length |S| = n. If h = h(S)n/q − 1, then h (S) = (S).
Conjecture 6.9 has been verified for cyclic groups of prime power order in [93]. The
following example shows that the conclusion of Conjecture 6.9 does not hold whenever
nq/(2n − q)h n/q − 2.
Let all notations be as in Conjecture 6.9, N = {0, a1 , . . . , an/q−1 } a subgroup of G with
|N| = n/q, g ∈ G with ord(g) = n and
h
W = a1h · . . . · an/q−1
g h (g + a1 )h · . . . · (g + an/q−1 )h ∈ F(G).
Since h nq/(2n − q), we have |W | = ( qn − 1)h + qn h n. Now let S be a subsequence of
W of length |S| = n such that g h (g + ai ) is a subsequence of S for some i ∈ [1, (n/q) − 1].
Then h(S) = h,
((h + 1)g + N ) ∩ (S) = ∅
but
((h + 1)g + N ) ∩ h (S) = ∅
whence h (S) = (S).
Next we discuss the invariants skn (G) where exp(G) = n and k ∈ N. If S ∈ F(G) is a
zero-sumfree sequence of length |S| = d(G) elements, then the sequence
T = 0kn−1 S
has no zero-sum subsequence of length kn whence skn (G)|T | + 1 = kn + d(G). The
following result may be found in [73].
Theorem 6.10. Let exp(G) = n2 and k ∈ N.
1. If k < D(G)/n, then skn (G) > kn + d(G).
2. If k |G|/n, then skn (G) = kn + d(G).
3. If G a finite abelian p-group and p l D(G), then spl k (G) = p l k + d(G).
Theorem 6.10 motivates the following definition.
Definition 6.11. We denote by l(G) the smallest integer l ∈ N such that
sk exp(G) (G) = k exp(G) + d(G) for every
k l.
Theorem 6.10 shows that
D(G)
|G|
l(G) n
n
whence l(Cn ) = 1.
Theorem 6.12. Let G = Cn1 Cn2 with 1 < n1 | n2 . Then l(G) = 2.
Proof. Since s(G) = 2n1 + 2n2 − 3 > n2 + d(G), it follows that l(G)2. Let k 2 and
S ∈ F(G) a sequence of length |S| = kn2 + d(G) = (k − 2)n2 + 3n2 + n1 − 2. We
prove that S has a zero-sum subsequence of length kn which implies that l(G) 2. Since
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W. Gao, A. Geroldinger / Expo. Math. 24 (2006) 337 – 369
s(G) = 2n1 + 2n2 − 3, S admits a product decomposition S = S1 · . . . · Sk−1 S where for all
i ∈ [1, k − 1], Si has sum zero and length |Si | = n2 (for details see [96, Lemma 5.7.10]).
Since |S |=|S|−(k +1)n2 =2n2 +n1 −1, Theorem 6.7.2.(b) implies that S has a zero-sum
subsequence Sk of length |Sk | ∈ {n2 , 2n2 } whence either S1 · . . . · Sk−1 Sk or S1 · . . . · Sk−2 Sk
is a zero-sum subsequence of length kn2 . The invariant Ek (G) was introduced in [72] (in connection with investigations on s(G),
see also [90]). Clearly, we have D(G)Ek (G) s(G), and if D(G) < k, then D(G)=Ek (G)
(see [157, Lemma 2.1]).
Theorem 6.13.
1. If G = Cn1 Cn2 with 1 n1 | n2 and n2 odd, then E2 (G) = 2n1 + 2n2 − 3 (see [72,
Section 3]).
2. If G = Cn Cn with n 2 and 3n, then E3 (G) = 3n − 2.
3. If G is a p-group and k ∈ N 2 with gcd(p, k) = 1, then
k
Ek (G) =
d∗ (G) + 1 (see [157]).
k−1
Proof. 2. By [157, Lemma 2.4], we have 3n − 2 E3 (G). Since snN (G) = 3n − 2, every
sequence S ∈ F(G) of length |S|3n − 2 has a zero-sum subsequence T of length
|T | ∈ {n, 2n} whence E3 (G) 3n − 2. 7. Inverse problems associated with (G) and s(G)
In this section we investigate the structure of sequences S ∈ F(G) of length
(G) − 1 without a zero-sum subsequence T of length |T | ∈ [1, exp(G)],
s(G) − 1 without a zero-sum subsequence T of length |T | = exp(G).
We formulate two properties and two conjectures.
Conjecture 7.1. Let S ∈ F(G) be a sequence of length |S| = s(G) − 1. If S has no
zero-sum subsequence of length exp(G), then h(S) = exp(G) − 1.
Note that Conjecture 7.1 and Fact F2 (formulated in the proof of Theorem 6.6) imply
Conjecture 6.5.
Property C. Every sequence S ∈ F(G) of length |S| = (G) − 1 which has no short
zero-sum subsequence has the form S = T n−1 for some sequence T ∈ F(G).
Property D. Every sequence S ∈ F(G) of length |S| = s(G) − 1 which has no zero-sum
subsequence of length n has the form S = T n−1 for some sequence T ∈ F(G).
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355
Suppose that G has Property D. We show that G satisfies Property C as well. Let S ∈
F(G) be a sequence of length (G) − 1 which has no short zero-sum subsequence. We
consider the sequence
T = 0n−1 S.
If T has a zero-sum subsequence T of length |T | = n, then T = 0k S with k ∈ [0, n − 1]
whence S is a short zero-sum subsequence of S, a contradiction. Thus T has no zero-sum
subsequence of length n. Since Property D holds, Conjectures 7.1 and 6.5 hold in G whence
|T |=(G)−1+(n−1)=s(G)−1. Therefore Property D implies that S has the required form.
Conjecture 7.2. Every group G = Cnr , where r ∈ N and n ∈ N 2 , has Property D.
An easy observation shows that
s(G)(g(G) − 1)(n − 1) + 1.
Moreover, if G = Cnr and equality holds, then Cnr has Property D (see [40, Lemma 2.3]).
Thus [119, Hilfssatz 3] implies that C3r has Property D for every r ∈ N. However, only
little is known for groups G = Cnr in case r 3 (see [91,82]).
We continue with some results on |G| (S) for general groups which arose from generalizations of the Erdős–Ginzburg–Ziv Theorem (see also [85,57,168,114] and note that
Theorem 7.3 implies Theorem 6.8). Then we discuss cyclic groups and groups of the form
G = Cn Cn .
Theorem 7.3 (See [60,61]). Let S ∈ F(G) be a sequence of length |S| |G| and let g ∈ G
with vg (S) = h(S).
1. |G| (S) = (|G|−h(S)) (−g + g −h(S) S).
2. Suppose that for every a ∈ G and every subsequence T of S of length |T | = |S| − |G| + 1
we have 0 ∈ (a + T ). Then
|G| (S) =
(y + S) = (−g + S).
y∈G
Next we present a result by D.J. Grynkiewicz [106, Theorem 1] which confirms a conjecture of Y. ould Hamidoune (see [116, Theorem 3.6] and [59] for special cases).
Theorem 7.4. Let S ∈ F(G) be a sequence of length |S| |G|+1, k ∈ N with |supp(S)| k
and h(S)|G| − k + 2. Then one of the following two statements holds:
(a) ||G| (S)| min{|G|, |S| − |G| + k − 1}.
(b) There exists a non-trivial subgroup H ⊂ G, some g ∈ G and a subsequence T of S
such that
the following conditions hold:
• H ⊂ |G| (S), |G| (S) is H -periodic and | |G| (S)| (|T | + 1)|H |.
, (G: H ) − 2}.
• supp(T −1 S) ⊂ g + H and |T | min{ |S−|G|+k−2
|H |
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Now we consider cyclic groups. Several authors [13,23,171,20,50] showed independently
that a sequence S ∈ F(Cn ) of length |S| = 2n − 2, which has no zero-sum subsequence
of length n, has the form S = a n−1 bn−1 where a, b ∈ Cn and ord(a − b) = n. Based on
Theorem 7.3 the following stronger result was obtained in [65, Theorem 1] (see also [88]).
Theorem 7.5. Let G be cyclic of order n 2, k ∈ [2, n/4 + 2] and S ∈ F(G) be a
sequence of length |S| = 2n − k. If S has no zero-sum subsequence of length n, then
S = a u b v c1 · . . . · c l ,
where ord(a − b) = n, u v n − 2k + 3 and
u + v 2n − 2k + 1 (equivalently, l k − 1). In particular, we have
• If k = 2, then S = a n−1 bn−1 .
• If k = 3 and n4, then S = a n−1 bn−2 or S = a n−1 bn−3 (2b − a).
Closely related to the inverse problem is the investigation of the Brakemeier function
(see [57,18,15,58,56,121,122]).
Conjecture 7.6. Let G be cyclic of order n2, q the smallest prime divisor of n and
S ∈ F(G) be a sequence of length |S|n + n/q − 1. If 0 ∈
/ n (S) then h(S)|S| − n + 1.
Conjecture 7.6 has been verified for cyclic groups of prime power order (see [92,93]).
The following example shows that the conclusion of Conjecture 7.6 does not hold whenever
q |S| − n n/q − 2.
Let all notations be as in Conjecture 7.6, N = {0, a1 , a2 , . . . , an/q−1 } be the subgroup of
G with |N| = n/q, k ∈ [q, n/q − 2], g ∈ G with ord(g) = n and
k
g k (g + a1 )k · . . . · (g + an/q−1 )k ∈ F(G)
W = a1k · . . . · an/q−1
a sequence of length |W | = k(2n/q − 1). Since k ∈ [q, n/q − 2], one can choose a
subsequence S of W such that |S| = n + k such that g k is a subsequence of S and (S) ∈
(k + 1)g + N. Therefore
h(S) = k and ((k + 1)g + N ) ∩ k (S) = ∅ which implies that
(S) ∈
/ k (S) and 0 ∈
/ n (S).
Now suppose that G = Cn Cn . It was P. van Emde Boas who studied Property C for
such groups in connection with his investigations on the Davenport constant for groups of
rank three (see [44] and [69, Lemma 4.7]). Property D was introduced in [70], where it is
shown that both Property C and Property D are multiplicative in the following sense.
Theorem 7.7. Let n1 , n2 ∈ N 2 . If the groups Cn1 Cn1 and Cn2 Cn2 both have Property C (or Property D, respectively), then the group Cn1 n2 Cn1 n2 has Property C
(or Property D, respectively).
The next result follows from Theorem 6.7.2.(b), from Theorem 7.7 and from [79,
Theorem 6.2].
Theorem 7.8. Let n 2 and suppose that n = m1 · . . . · ms where s ∈ N and m1 , . . . , ms ∈
N 2 . If for all i ∈ [1, s] the groups Cmi Cmi satisfy the equivalent conditions of Theorem
4.4, then Cn Cn has Property C.
W. Gao, A. Geroldinger / Expo. Math. 24 (2006) 337 – 369
357
In [125] it is shown that Cp Cp has Property D for p ∈ {2, 3, 5} and in [165] the same
is shown for p = 7. We end with a result which could be a first step on the way showing
that Cn Cn has Property C.
Theorem 7.9. Let G = Cn Cn with n 3 and S = f1n−1 f2n−1 g1 · . . . · gn−1 ∈ F(G) be
a sequence of length |S| = 3n − 3 which has no short zero-sum subsequence. Then there
exists a basis (e1 , e2 ) of G such that
S = (e1 + e2 )n−1 e2n−1
n−1
(ai e1 + be2 ),
i=1
where ai ∈ [0, n − 1] for all i ∈ [1, n − 1] and b ∈ [0, n − 1]\{1}.
Proof. By [96, Lemma 5.8.6] it follows that (f1 , f2 ) is a basis of G whence gi =yi f1 +xi f2
with xi , yi ∈ [0, n − 1] for all i ∈ [1, n − 1]. We assert that
x1 + y1 = · · · = xn−1 + yn−1 .
Assume to the contrary that this does not hold. Then Theorem 4.2.2 implies that the sequence
n−1
((xi + yi − 1)e1 )
is not zero-sumfree.
i=1
Hence after some renumeration we may suppose that
t
(xi + yi − 1) ≡ 0 mod n
for some t ∈ [1, n − 1].
i=1
Then the sequence
n−y
W = f2n−x f1
t
(yi f1 + xi f2 ),
i=1
where x, y ∈ [1, n] such that x ≡ x1 + · · · + xt mod n and y ≡ y1 + · · · + yt mod n, is
a zero-sum subsequence of S of length |W | = (n − x) + (n − y) + t ≡ 0 mod n. Since S
has no short zero-sum subsequence, it follows that |W | = 2n. But then |W | > d(Cn Cn )
whence W (and thus S) has a short zero-sum subsequence, a contradiction.
Now we obtain that (e1 , e2 ) = (f2 − f1 , f1 ) is a basis of G and
gi = yi f1 + xi f2 = xi e1 + (xi + yi )e2
for all i ∈ [1, n − 1].
/ 1 mod n. Assume to the contrary that (x1 + y1 )e2 =
Thus it remains to show that x1 + y1 ≡
e2 . Since s(Cn ) = 2n − 1, the sequence e1n−1 0n−1 (x1 e1 ) has a zero-sum subsequence of
length n whence (e1 + e2 )n−1 e2n−1 (x1 e1 + e2 ) has a zero-sum subsequence of length n,
a contradiction. 358
W. Gao, A. Geroldinger / Expo. Math. 24 (2006) 337 – 369
8. On the number of zero-sum subsequences
The enumeration of zero-sum subsequences of a given (long) sequence over G, which
have some prescribed properties, is a classical topic in combinatorial number theory going
back to P. Erdős, J.E. Olson and others. Many zero-sum results (such as the proof of d∗ (G)=
d(G) for p-groups or the proof that s(Cp Cp ) = 4p − 3) are based on enumeration results.
Definition 8.1. Let S = g1 · . . . · gl ∈ F(G) be a sequence of length |S| = l ∈ N0 and
let g ∈ G.
1. For every k ∈ N0 let
k
Ng (S) = I ⊂ [1, l]|
gi = g and |I | = k i∈I
denote the number of subsequences T of S having sum (T ) = g and length |T | = k
(counted with the multiplicity of their appearance in S). In particular, N00 (S) = 1 and
N0g (S) = 0 if g ∈ G• .
2. We define
−
Ng (S) =
Nkg (S), N+
N2k
N2k+1
(S).
g (S) =
g (S) and Ng (S) =
g
k 0
k 0
k 0
Thus Ng (S) denotes the number of subsequences T of S having sum (T ) = g, N+
g (S)
−
denotes the number of all such subsequences of even length, and Ng (S) denotes the
number of all such subsequences of odd length (each counted with the multiplicity of
its appearance in S).
We start with two results on p-groups. The first one (see [75]) sharpens results of
J.E. Olson and I. Koutis (see [143, Theorem 1] and [128, Theorems 7–10]). It is proved via
group algebras.
Theorem 8.2. Let G be a p-group, g ∈ G, k ∈ N0 and S ∈ F(G) be a sequence of length
|S| > k exp(G) + d∗ (G).
−
k+1 .
1. N+
g (S) ≡ Ng (S) mod p
2. If p = 2, then Ng (S) ≡ 0 mod 2k+1 .
The next result (proved in [73]) is based on Theorem 8.2.
Theorem 8.3. Let G be a p-group and S ∈ F(G) be a sequence of length |S| ∈ [|G| +
d(G), 2|G| − 1]. Then
0 mod p if g ∈ G• ,
|G|
Ng (S) ≡
1 mod p if g = 0.
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359
An easy argument shows that in an elementary 2-group we have N0 (S) = Ng (S) for every
S ∈ F(G) and every g ∈ (S) (see [75, Proposition 3.3]). For more enumeration results
in G = Cp see [64], and in G = Cp Cp see [96, Theorems 5.8.1 and 5.8.2].
We continue with some results of the following type: A sequence S ∈ F(G), for which
|S| is long and |(S)| is small, has a very special form. The first result is due to J.E. Olson
[149, Theorems 1 and 2].
Theorem 8.4. Let S ∈ F(G• ) be a sequence of length |S| = |G|. If N0 (S) < |G|, then G
is cyclic and S = g |G| for some g ∈ G• .
For cyclic groups there are the following two sharper results: for Theorem 8.5 see [67,
Theorem 1] (note that there is a misprint in the formulation of Theorem 1), and for Theorem
8.6 see [67, Theorems 2–4].
Theorem 8.5. Let G be cyclic of order n 2, k ∈ [1, n/4 + 1] and S ∈ F(G).
If N0 (S) < 2|S|−n+k+1 , then there exists some g ∈ G with ord(g) = n such that
S = g u (−g)v (x1 g) · . . . · (xk−1 g)(y1 g) · . . . · (yl g),
where uv 0, u + v
= n − 2k + 1,
yi ∈ [0, n − 1] for all i ∈ [1, l], xi ∈ [1, n − 1] for
all i ∈ [1, k − 1] and xi n/2 xi + xi >n/2 (n − xi ) 2k − 2.
Theorem 8.6. Let G be cyclic of order n22 and S ∈ F(G• ) be a sequence of length
|S| = n − 1. If N0 (S) n, then there exists some g ∈ G with ord(g) = n such that S has one
of the following forms:
(−g)g n−2 , (2g)(−g)g n−3 , (3g)(−g)g n−3 , (2g)2 (−g)g n−4 , g n−1 , (2g)g n−2 ,
(3g)g n−2 , (2g)2 g n−3 .
The next result deals with the number of zero-sum subsequences of length exp(G) in
cyclic groups (see [68]).
Theorem 8.7. Let G be cyclic of order n 2 and S ∈ F(G) be a sequence of length
|S| = 2n − 1.
1. For every g ∈ G• we have Nng (S) = 0 or Nng (S) n.
2. Nn0 (S)n + 1 or S = a n bn−1 for some a, b ∈ G with ord(a − b) = n.
The following examples show that the inequalities in Theorem 8.7 cannot be improved.
Let g ∈ G with ord(g) = n. If
S = 0n−1 g n−1 (−g), then Nn−g (S) = n,
and if
S = 0n+1 g n−2 , then Nn0 (S) = n + 1.
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W. Gao, A. Geroldinger / Expo. Math. 24 (2006) 337 – 369
A problem related to Theorem 8.7 on Nn0 (S) is the following conjecture formulated by
A. Bialostocki and M. Lotspeich [126,55]:
Conjecture 8.8. Let G be cyclic of order n 2 and S ∈ F(G). Then
|S|/2
|S|/2
+
.
Nn0 (S)
n
n
Z. Füredi and D. Kleitman, M. Kisin, W. Gao and D.J. Grynkiewicz gave partial positive
answers to the above conjecture.
Theorem 8.9. Conjecture 8.8 holds true in each of the following cases:
1. n = p a q b where p, q are distinct primes, a ∈ N and b ∈ {0, 1} (see [55]).
2. |S| n6n (see [126]).
3. |S|19n/3 (see [102], and also [68]).
The next result (see [84]) settles a conjecture of B. Bollobás and I. Leader (see [16]).
Theorem 8.10. Let S ∈ F(G) be a sequence. If 0 ∈
/ |G| (S), then there is a zero-sumfree
sequence T ∈ F(G) of length |T | = |S| − |G| + 1 such that ||G| (S)| |(T )|.
We conclude with an explicit formula for the number of all zero-sum sequences of given
length, which was recently derived by V. Ponomarenko (see [154]).
Theorem 8.11. Let G be cyclic of order n10 and k > 2n/3. Then
|{S ∈ A(G)||S| = k}| = (n)pk (n),
where is Euler’s Phi Function and pk (n) denotes the number of partitions of n into k
parts.
9. Weighted sequences and the cross-number
We start with a recent result due to D.J. Grynkiewicz (see [104, Theorem 1.1]) which
may be considered as a weighted version of the Theorem of Erdős–Ginzburg–Ziv (the case
where G is cyclic, k = |G| and w1 = · · · = wk = 1 gives the classical result). It completely
affirms a conjecture of Y. Caro formulated in 1996 (see [23, Conjecture 2.4]). Special
cases were settled by N. Alon et al. [23], by W. Gao and X. Jin [83] and byY. ould Hamidoune
[113, Theorem 2.1].
Theorem 9.1. Let S ∈ F(G) be a sequence of length |S| = |G| + k − 1, for some k 2,
and (w1 , . . . , wk ) ∈ Zk a k-tuple of integers such that w1 + · · · + wk ≡ 0 mod exp(G).
Then S has a subsequence T = g1 · . . . · gk such that w1 g1 + · · · + wk gk = 0.
We continue with a result by Y. ould Hamidoune [113, Theorem 3.2] which implies
Theorem 6.8 (for results of a similar flavor see [34,111,117]).
W. Gao, A. Geroldinger / Expo. Math. 24 (2006) 337 – 369
361
Theorem 9.2. Let S ∈ F(G) be a sequence of length |S| = D(G) + k with k |G| − 1
and let g ∈ G with vg (S) = h(S). Then S has a subsequence T of length |T | = k such that
(T ) = kg.
Next we discuss the cross number of a finite abelian group. It was introduced by U. Krause
(see [129,130]), and its relevance stems from the theory of non-unique factorizations (see
[158] and [96, Chapter 6]).
Definition 9.3. The invariant
K(G) = max{k(S)|S ∈ A(G)}
is called the cross-number of G and
k(G) = max{k(S)|S ∈ F(G) is zero-sumfree}
is called the little cross-number of G.
If exp(G) = n and q is the smallest prime divisor of n, then a straightforward argument
(see [96, Proposition 5.1.8]) shows that
1
1
1
+ k∗ (G) + k(G) K(G) + k(G).
n
n
q
Conjecture 9.4.
1
n
+ k∗ (G) = K(G).
Conjecture 9.4 has been verified for p-groups and various other classes of groups (see
[96, Theorem 5.5.9 and Section 5.7]).
Theorem 9.5.
1. 1 + nk(G) is the smallest integer l ∈ N such that every sequence S ∈ F(G) with
nk(S)l has a non-empty zero-sum subsequence.
2. Every sequence S ∈ F(G) of length |S| |G| has a non-empty zero-sum subsequence
T with k(T )1.
Whereas Theorem 9.5.1 is straightforward, Theorem 9.5.2 settles a conjecture of D.
Kleitman and P. Lemke (see [127,95] and [42] for a recent graph theoretical approach). For
more information on the cross number we refer to [99,28,100,101,7].
10. On the Olson constant, the critical number and some analogues
We summarize some basic relationships of the invariants introduced in Definition 2.2.
Note that max{|U ||U ∈ A(G) squarefree} is called the strong Davenport constant of G
(see [26,51,27,151]).
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W. Gao, A. Geroldinger / Expo. Math. 24 (2006) 337 – 369
Lemma 10.1.
1.
2.
3.
4.
5.
1 + ol(G) = Ol(G) g(G) |G| + 1.
g(G) = |G| + 1 if and only if G is either cyclic of even order or an elementary 2-group.
1 + max{|S||S ∈ F(G) squarefree, (S) = G• } Ol(G) min{D(G), cr(G)}.
max{|supp(U )||U ∈ A(G)} = max{|U | | U ∈ A(G) squarefree} Ol(G).
√
If f(G, l)1 + c−2 l 2 for some l ∈ N and c ∈ R>0 , then ol(G) < c |G| − 1.
Proof. We show the upper bound on g(G) and 2. A proof of 4. may be found in
[26, Theorem 7], and the remaining assertions follow either by the very definitions or
by [96, Lemma 5.1.17]. Since there are no squarefree sequences S ∈ F(G) of length
|S||G| + 1, every such sequence has a zero-sum subsequence of length |T | = exp(G)
whence g(G)|G|+1. If G is cyclic of even order or an elementary 2-group, then the squarefree sequence S ∈ F(G) consisting of all group elements has no zero-sum subsequence T
of length |T | = exp(G) whence g(G) > |G|. Suppose that G = H g with some (possibly
trivial) subgroup H ⊂ G and some g ∈ G with ord(g) = exp(G) = n 3. We have to show
that the squarefree sequence S ∈ F(G) consisting of all group elements as a zero-sum subsequence T of length |T |=n. If n is odd, then T =g(2g)·. . .·(ng) has the required property.
If n is even and h ∈ H \{0}, then T = g(2g) · . . . · ((n − 2)g)(h + (n − 1)g)(−h + (n/2)g)
has the required property. We start with the g-invariant which was first studied by H. Harborth and A. Kemnitz (see
[119,125]). Let G = Cn Cn with n 3 and let (e1 , e2 ) be a basis of G. If n is odd, then
S=
n−2
(ie2 )
i=0
n−1
(e1 + ie2 ) ∈ F(G)
i=1
is a squarefree sequence of length |S| = 2n − 2 which has no zero-sum subsequence of
length n whence g(G) 2n − 1. If n is even, then
S=
n−1
i=0
(ie2 )
n−1
(e1 + ie2 ) ∈ F(G)
i=0
is a squarefree sequence of length |S| = 2n which has no zero-sum subsequence of length
n whence g(G)2n + 1.
Conjecture 10.2. Let G = Cn Cn with n 3. Then
2n − 1 if n is odd,
g(G) =
2n + 1 if n is even.
Conjecture 10.2 holds true for some small integers and for all primes p 67 (see [92]).
We continue with the Olson constant. For some basic bounds for the f-invariant (and
hence for the Olson constant) we refer to [96, Section 5.3]. Proving a conjecture of P. Erdős
and H. Heilbronn, E. Szemerédi [166] showed that there is some c ∈ R>0 (not depending
W. Gao, A. Geroldinger / Expo. Math. 24 (2006) 337 – 369
363
√
on the group) such that Ol(G)c |G|. J.E. Olson [145] proved the result for c = 3. The
following result is due to Y. ould Hamidoune and G. Zémor [118, Theorems 3.3 and 4.5].
Theorem 10.3.
√
1. If G is prime
√ cyclic, then Ol(G) 2|G| + 5 log(|G|).
2. Ol(G) 2|G| + ε(|G|) for some real-valued function ε with ε(x) = O(x 1/3 log x).
The result for prime cyclic groups is essentially the best possible. However, the situation
is completely different for non-cyclic groups. We have ol(G)d(G), and obviously equality
holds for elementary 2-groups, and by [163] also for elementary 3-groups. In the following
theorem we summarize two results. The first one (see [77, Theorem 7.3]) shows in particular
that in p-groups of large rank we have ol(G) = d(G) (which is in contrast to the situation
in Cp Cp , see Theorem 4.4). The second result was recently achieved in [89].
Theorem 10.4.
1. Let G = H Cns+1 where exp(G) = n 2, s ∈ N0 , H ⊂ G a (possibly trivial) subgroup
and exp(H ) a proper divisor of n. If r(H ) + s/2 n, then 1 + d∗ (G) max{|U ||U ∈
A(G) squarefree}.
2. Ol(Cp Cp ) = Ol(Cp ) + p − 1 for all primes p > 4 · 67 × 1034 .
Let G = H Cn = H e where H ⊂ G is a subgroup with |H | n − 1 and e ∈ G
with ord(e) = n. If T ∈ F(H ) is a squarefree zero-sumfree sequence of length |T | = ol(G)
and h1 , . . . , hn−1 ∈ H are pairwise distinct, then
S=T
n−1
(e + hi ) ∈ F(G)
i=1
is a squarefree zero-sumfree sequence of length |S| = |T | + n − 1 whence ol(G) ol(H ) +
n − 1. Let n be a prime power. Assume to the contrary that Ol(Cnr ) = Ol(Cnr−1 ) + n − 1 for
all r 2. Then Theorem 10.4.1 implies that Ol(Cn ) = D(Cn ), a contradiction. Thus there
exists some r 2 such that Ol(Cnr ) > Ol(Cnr−1 ) + n − 1.
Finally we discuss the critical number cr(G) of G. It was first studied by P. Erdős and
H. Heilbronn (see [48, Theorem I]) for cyclic groups of prime order, and in the sequel this
problem found a lot of attention (see [138,38,37,139,153,30,33,134,86,115]). Following
[87] (where the inverse problem associated to the critical number is studied) we summarize
what is known on cr(G).
Theorem 10.5. Let q denote the smallest prime divisor of exp(G).
√
1. Suppose that |G| = q. Then cr(G) 4q − 7, and equality holds if the upper bound
is odd (see [33, Example 4.2]).
2. Suppose that |G|/q is prime.
(a) cr(C2 C2 ) = 3, and if q is odd, then cr(Cq Cq ) = 2q − 2.
(b) |G|/q + q − 2 cr(G)|G|/q + q − 1.
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W. Gao, A. Geroldinger / Expo. Math. 24 (2006) 337 – 369
3. Suppose that |G|/q is composite. We have cr(C8 ) = cr(C2 C4 ) = 5, and otherwise
cr(G) =
|G|
+ q − 2.
q
C. Peng (see [153,152,66]) investigated the following variant of the critical number. He
studied the smallest integer l ∈ N0 with the following property: every sequence S ∈ F(G• )
of length |S|l and with |supp(S) ∩ H ||H | − 1 for all proper subgroups H ⊂ G, satisfies
(S) = G.
Van H. Vu (see [169]) showed the existence of a constant C with the following property: If
G is a sufficiently large cyclic group√and S ∈ F(G) a squarefree sequence with supp(S) ⊂
{g ∈ G | ord(g) = |G|} and |S|C |G|, then (S) = G• .
Acknowledgments
The first author is supported by NSFC, Project No. 10271080. The second author is
supported by the Austrian Science Fund FWF, Project No. P18779-N13.
Note added in proof
When this article went to press in June 2006, we were informed on the following progress:
• S. Savchev and F. Chen announced an improvement of Theorem 4.2.
• D.J. Grynkiewiecz, O. Ordaz, M.T. Varela and F. Villarroel announced progress on Conjectures 6.9 and 7.6.
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