Abstract:
A conjecture by Erdős and Lemke in elementary number theory goes as follows: If d is
a divisor of n and we have d divisors of n, say a1,...,ad, not necessarily distinct, can we
always find a subsequence among them such that their sum is (i) divisible by d, and (ii) at
most n? -This was proved by Lemke and Kleitman to be indeed the case. They also noted
that an equivalent version of their theorem, stated in terms of the additive cyclic group
G = Zn is as follows: Every sequence of n elements of G, not necessarily distinct, contains
a subsequence g1,...,gk such that g1+...+gk = 0 and
Σki=1/|gi|<=1. This has been shown to be correct for every nite abelian group G. Hence a natural question is therefore if this holds true for any finite group G. By the aid of a computer this has been verified
for all solvable groups of order 21 or less, but it is still not known whether it holds for
all finite groups. - This paper proves that some well-known non-abelian groups have this property, for example the alternating groups An and symmetric groups Sn for n = 3,4,5,6, the dihedral group Dn for every n and the dicyclic group of every order. Some speculations
on possible plan of attack for Sn for larger n are finally discussed.
