Zero Sum Properties in Groups

dc.contributor.advisorAgnarsson, Geir
dc.contributor.authorWu, Angelina A.
dc.creatorWu, Angelina A.
dc.description.abstractA 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.
dc.subjectZero sum
dc.subjectSymmetric group
dc.subjectGroup partition
dc.subjectDavenport Constant
dc.titleZero Sum Properties in Groups
dc.typeThesis Mason University's of Science in Mathematics


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