Research on relative difference sets of affine type

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We study relative difference sets in the simple group Alt (5) and show that up to equivalence there exist exactly two nontrivial relative difference sets in Alt (5), which are of affine type. It is conjectured that there exists no nontrivial ordinary difference set in dihedral groups. In this connection we also study relative difference sets in dihedral groups and show that there is no nontrivial relative difference set of semiregular type or of affine type in them. Concerning semiregular relative difference sets S. L. Ma and B. Schmidt conjectured that the only semiregular relative difference set in Zp^2 × Zp^2 is a trivial one. Using characters of Zp^2 × Zp^2 we prove that the conjecture is true when p>3. Therefore it is conceivable that there exists no nontrivial semiregular relative difference set in Zp^n ×Zp^n for any n>1. Moreover, we study relation between affine difference sets and group extensions. Let D be an abelian affine difference set of odd order n in a group G of order n2-l with respect to a subgroup N of order n-1. Set H=G/N={t o, t n). Then we can select a suitable factor set f: H×H N such that g (σ)=&.σ, to)f (σ, t i)-f (σ, tn) satisfies (1) g (σm)=g (σ)m for any divisor m of n and (2) g (σ)=g (σ1) for any a in H. Using these nice properties we prove that the multiplicative order of m mod exp (N) is at most 2 times the multiplicative order of m mod exp (H). As an application we know that n is a power of a prime if n ≦100000 with 155 exceptions