The structure of the exponent set for finite cyclic groups

Details The structure of the exponent set for finite cyclic groups The structure of the exponent set for finite cyclic groups

2008-10-06

arxiv.org/abs/0810.0881v2

Mathematics

Number Theory

This paper has been withdrawn since it's primary content is now subsumed by new work of the authors and Peter Hegarty

Scientific paper

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 $n$.

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