# Zero Sum Properties in Groups

 dc.contributor.advisor Agnarsson, Geir dc.contributor.author Wu, Angelina A. dc.creator Wu, Angelina A. dc.date 2013-12-02 dc.date.accessioned 2014-03-09T14:15:41Z dc.date.available 2014-03-09T14:15:41Z dc.date.issued 2014-03-09 dc.description.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. dc.identifier.uri https://hdl.handle.net/1920/8651 dc.language.iso en dc.subject Zero sum dc.subject Isomorphism dc.subject Symmetric group dc.subject Group partition dc.subject Davenport Constant dc.title Zero Sum Properties in Groups dc.type Thesis thesis.degree.discipline Mathematics thesis.degree.grantor George Mason University thesis.degree.level Master's thesis.degree.name Master of Science in Mathematics

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