The Structure of the Exponent Set for Finite Cyclic Groups

We survey properties of the set of possible exponents of subsets of Zn (equivalently, exponents of primitive circulant digraphs on n vertices). Let En denote this exponent set. We point out that En contains the positive integers up to √ n, the ‘large’ exponents ⌊ 3 ⌋+1, ⌊ 2 ⌋, n−1, and for even n ≥ 4, the additional value n 2 −1. It is easy to see that no exponent in [n 2 +1, n−2] is possible, and Wang and Meng have shown that no exponent in [⌊ 3 ⌋+2, n 2 −2] is possible. Extending this result, we show that the interval [⌊ 4 ⌋+3, ⌊ 3 ⌋− 2] is another gap in the exponent set En. In particular, 11 6∈ E35 and this gap is nonempty for all n ≥ 57. A conjecture is made about further gaps in En for large n.