# Olson's Conjecture

 Importance: Medium ✭✭
 Author(s): Olson, John E.
 Subject: Number Theory » Combinatorial Number Theory
 Keywords: zero sum
 Recomm. for undergrads: no
 Prize: none
 Posted by: mdevos on: March 10th, 2007
Conjecture   If is a sequence of elements from a multiplicative group of order , then there exist so that .

A famous theorem of Erdos, Ginzburg, and Ziv asserts that every sequence of elements from an additive abelian group has a subsequence of length which sums to . This pretty result has lead to numerous generalizations. In particular, Olsen generalized this result by showing that every sequence of elements from an arbitrary multiplicative group of order has a subsequence of length which has product equal to in some order. The above conjecture asserts that this reordering is not needed. Apart from Olson's result, there appears to be very little known about this problem. Next we highlight an obvious question which appears untouched.

For every finite multiplicative group , let denote the smallest integer so that every sequence of elements of has a subsequence of length with product equal to in the given order (so Olsen's conjecture is equivalent to ). It is clear that , since any sequence of length must contain at least copies of the same element, and the product of these will be . However, I (M. DeVos) don't know how to improve significantly on this upper bound, and it would appear to me that any significant progress in this direction would require a little something new.