An Exponent Bound for Relative Difference Sets in p-Groups

An exponent bound ia presented for abelian (pi+j, p0, p•+;, pi) relative c:liffere.nce sets: this botllld can be met for i $ j. RESULT For background in Relative Difference Sets (RDS), see [3]. The basic group ring equation for a RDS Din a group G with a forbidden subgroup N is nn<-1> = k + A(GN) If x is a character on the abelian group G, then we have three possibilities for I x(D) l=I L:dED x(d) I: if x is the principal character (identically 1) on G, then x(D) = k. If xis principal on N but nonprincipal on G, then I x(D) I= Jk A IN I· Finally, if x is a nonprincipal character on N , then I x(D) I= ../f. The last possibility is the case considered in this paper. We will consider the following parameters: v = p2;+;, I N I= p1, k = p;+;, and A = pi. Many examples of RDS with these parameters can be found in [l],[2],and [3). If Dis a RDS with these parameters, and x is a character that is nonprincipal on N, then I x(D) I= pi¥. Consider the i + j even case. This last equation transfers a group ring question into a number theoretic question; namely, when can the algebraic integer x(D) have modulus p!ti . This question was considered in the classical paper by Turyn [4). He based many of the arguments in that paper on the result due to Kronecker that if A and B are algebraic integers in the number field Q[{) ({ a n1h root of unity), and (A)= (B) as ideals, then A= B{i for some j (see p.321 of [4]). This implies that x(D) =pi¥ e; fore a '[I' root of unity. If we rewrite x(D) = E~=l Y;{i' then all of the Yi will be 0 except one, which will be p.!¥. Since x is a homomorphism of G, we can bound the Y; by 0 $ Yi $I Ker(x) I· Thus, I Ker(x) I has to be at least p.!¥ in order for the character sum to work. If we define the exponent of the group (written ezp(G)) is the size of the largest cyclic subgroup, and the order of x is the smallest n so that (x(g)r = 1 for every g E G, then there is a character x of ARS COMBINATORIA 34(1992), pp. 318-320