View source for Davenport constant

<!--
d1100101.png
$#A+1 = 29 n = 0
$#C+1 = 29 : ~/encyclopedia/old_files/data/D110/D.1100010 Davenport constant
Automatically converted into TeX, above some diagnostics.
Please remove this comment and the {{TEX|auto}} line below,
if TeX found to be correct.
-->

{{TEX|auto}}
{{TEX|done}}

For a finite [[Abelian group|Abelian group]]  $  G $, 
Davenport's constant  $  D ( G ) $
of  $  G $
is the smallest positive integer  $  d $
such that for any sequence  $  S = ( g _ {1} \dots g _ {d} ) $
of not necessarily distinct elements of  $  G $
there is a non-empty subsequence  $  S  ^  \prime  \subseteq S $
with sum zero (i.e., there is some  $  \emptyset \neq I \subseteq \{ 1 \dots d \} $
with  $  \sum _ {i \in I }  g _ {i} = 0 $).

It is reported in [[#References|[a7]]] that in 1966 H. Davenport proposed the problem of finding  $  D ( G ) $
in the following connection. Let  $  K $
be an algebraic number field (cf. also [[Algebraic number|Algebraic number]]; [[Field|Field]]) with ring of integers  $  R $
and ideal class group  $  G $. 
Then  $  D ( G ) $
is the maximal number of prime ideals (counted with multiplicity) which can divide an irreducible element of  $  R $. 
This is the reason why  $  D ( G ) $
is a crucial invariant in the theory of non-unique factorizations [[#References|[a2]]].

Let  $  C _ {n} $
denote the [[Cyclic group|cyclic group]] with  $  n $
elements and suppose  $  G = C _ {n _ {1}  } \oplus \dots \oplus C _ {n _ {r}  } $
with  $  1 < n _ {1} | \dots | n _ {r} $
and with rank  $  r ( G ) = r $. 
Then

$$ 
1 + \sum _ {i = 1 } ^ { r }  ( n _ {i} - 1 ) \leq  D ( G ) \leq  n _ {r} \left ( 1 + { \mathop{\rm log} } {
\frac{\left | G \right | }{n _ {r} }
 } \right ) .
$$

In the left-hand inequality, equality holds for  $  p $-
groups and for groups  $  G $
with  $  r ( G ) \leq  2 $(
proved independently by J.E. Olson [[#References|[a7]]] and D. Kruyswijk [[#References|[a3]]]; cf. also [[P-group| $  p $-
group]]). However, the left-hand inequality can be strict [[#References|[a4]]], [[#References|[a5]]]. For the right-hand inequality, cf. [[#References|[a6]]]. It is still (1996) an open question whether the left-hand inequality can be strict for groups of rank  $  3 $
or for groups of the form  $  G = C _ {n}  ^ {r} $.

The problem of determining  $  D ( G ) $
belongs to the area of zero-sum sequences, a subfield of [[Additive number theory|additive number theory]], respectively additive [[Group|group]] theory. For related problems cf. [[#References|[a1]]] and the literature cited there.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Alon,   M. Dubiner,   "Zero-sum sets of prescribed size"  D. Miklós (ed.)  V.T. Sós (ed.)  T. Szönyi (ed.) , ''Combinatorics, Paul Erdös is Eighty'' , ''Bolyai Society Mathematical Studies'' , '''1''' , Keszthely (Hungary)  (1993)  pp. 33–50</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S. Chapman,   "On the Davenport's constant, the cross number and their application in factorization theory" , ''Lecture Notes in Pure and Appl. Math.'' , '''171''' , M. Dekker  (1995)  pp. 167–190</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  P. van Emde Boas,   D. Kruyswijk,   "A combinatorial problem on finite abelian groups III"  ''Report Math. Centre'' , '''ZW–1969–008'''  (1969)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  A. Geroldinger,   R. Schneider,   "On Davenport's constant"  ''J. Combin. Th. A'' , '''61'''  (1992)  pp. 147–152</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  M. Mazur,   "A note on the growth of Davenport's constant"  ''Manuscr. Math.'' , '''74'''  (1992)  pp. 229–235</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  R. Meshulam,   "An uncertainty inequality and zero subsums"  ''Discrete Math.'' , '''84'''  (1990)  pp. 197–200</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  J.E. Olson,   "A combinatorial problem on finite abelian groups I–II"  ''J. Number Th.'' , '''1'''  (1969)  pp. 8–10; 195–199</TD></TR></table>