The structure of the exponent set for finite cyclic groups

TL;DR

This paper surveys the set of possible exponents of subsets of cyclic groups, identifying known values, gaps, and extending previous results to find new intervals where exponents cannot occur, and conjectures further gaps for large n.

Contribution

It extends previous work by identifying new gaps in the exponent set for cyclic groups and proposes conjectures about additional gaps for large n.

Findings

Identifies known exponents in E_n.

Shows specific intervals are gaps in E_n.

Proposes conjecture on further gaps for large n.

Abstract

We survey properties of the set of possible exponents of subsets of $\Z_n$ (equivalently, exponents of primitive circulant digraphs on $n$ vertices). Let $E_n$ denote this exponent set. We point out that $E_n$ contains the positive integers up to $\sqrt{n}$, the `large' exponents $\lfloor \frac{n}{3} \rfloor +1, \lfloor \frac{n}{2} \rfloor, n-1$, and for even $n \ge 4$, the additional value $\frac{n}{2}-1$. It is easy to see that no exponent in $[\frac{n}{2}+1,n-2]$ is possible, and Wang and Meng have shown that no exponent in $[\lfloor \frac{n}{3}\rfloor +2,\frac{n}{2}-2]$ is possible. Extending this result, we show that the interval $[\lfloor \frac{n}{4} \rfloor +3, \lfloor \frac{n}{3} \rfloor -2]$ is another gap in the exponent set $E_n$. In particular, $11 \not\in E_{35}$ and this gap is nonempty for all $n \ge 57$. A conjecture is made about further gaps in $E_n$ for large $n$.