Daisy chains with three generators

For many positive odd integers n, whether prime, prime power or composite, the set U-n of units of Zn contains members u, v and w, say with respective orders psi, omega and Pi, such that we can write Un as the direct product U-n = < u > x < v > x < w >. Each element of Un can then be written in the form u(i) v(j) w(k) where 0 <= i <= psi - 1, 0 <= j <= w - 1 and 0 <= k <= pi - 1. We can then often use the structure of < u > x < v > x < w > to arrange the psi omega pi elements of U-n in a daisy chain, i.e. in a circular arrangement such that, as we proceed round the chain in either direction, the set of differences between each member and the preceding one is itself the set U-n. We describe such daisy chains as daisy chains with three generators. Each such daisy chain consists of a succession of super-segments of length omega pi, each made of segments of length pi. Within each segment, each successive element is obtained from the preceding one by multiplication by w; within each super-segment, each successive segment is obtained from the preceding one by multiplication by v; each successive super-segment is obtained from the preceding one by multiplication by u. We study the existence of such arrangements, some of which can be obtained from general constructions which we describe. In many of our examples of the arrangements, one of the generators has order 2; if n is prime, that generator must then be - 1 (mod n), but if n is composite, another square root of 1 (mod n) may occasionally be used.