A note on the Multiplier Conjecture |
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Authors: | Gerhard Gerlich |
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Institution: | 1. Institut für Geometrie, Algebra und diskrete Mathematik, Technische Universit?t Braunschweig, Pockelsstr. 14, 38106, Braunschweig, Germany
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Abstract: | In order to identify multipliers of abelian (υ, k, λ)-difference sets the First and the Second Multiplier Theorem of Hall, Ryser and Chowla, resp. of Hall and Menon, need
a divisor m of n = k − λ that is coprime to υ. Moreover, both theorems require that m > λ. The famous Multiplier Conjecture asserts that the restriction m > λ is not necessary.
We present a generalization of the Second Multiplier Theorem where m is not necessarily coprime to υ. Here the requirement that m > λ generalizes to the condition m/(υ, m) > λ. This gives rise to a generalized Multiplier Conjecture which asserts that this condition is not necessary. We disprove
this conjecture by showing that there exist counterexamples. |
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Keywords: | 05B10 |
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