This paper was initiated by the question whether any non-abelian difference set exists without having a single abelian difference set sharing the same parametersr-K. Smith has found the first example of such a non-abelian difference set. We will study the possibility of abelianizing non-abelian difference sets. We will prove some coexistence results about certain types of non-abelian and abelian difference sets. As a result we can prove, for an infinite series of parameters, that several classes of non-abelian difference sets do not exist.
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