Research on relative difference sets of affine type

研究了单群Alt(5)中的相对差集，证明了在等价条件下，Alt(5)中恰好存在两个仿射类型的非平凡相对差集。猜想二面体群中不存在非平凡的普通差集。在这方面，我们还研究了二面体群中的相对差集，并证明了它们中不存在半正则型或仿射型的非平凡相对差集。关于半正则相对差集，S.L.Ma和B.Schmidt猜想ZP^2×ZP^2中唯一的半正则相对差集是平凡的。利用Zp^2×Zp^2的性质，证明了当p&gt；3时，猜想成立，从而可想而知，对于任意n&gt；1，Zp^n×Zp^n中不存在非平凡的半正则相对差集。此外，我们还研究了仿射差集与群扩张之间的关系。设D是关于n-1阶子群N的n2-L阶群G中的n阶奇数仿射差集。集合H=G/N={to，tn)。然后我们可以选择一个合适的因子集f：H×H N使得g(σ)=&.σ，to)f(σ，ti)-f(σ，tn)满足(1)g(σm)=g(σ)m，(2)g(σ)=g(σ1)对于H中的任何a。利用这些性质，我们证明了m mod exp(N)的乘法阶至多是m mod exp(H)的乘法阶的2倍。作为一个应用，我们知道n是素数的幂，如果n≦100000，则有155个例外

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