Rotary pairs in symmetric groups of composite degree

Find all rotary pairs (a,z) for the symmetric group S_n when n>5.

Background

The paper studies rotary pairs in symmetric groups and proves a counting result when the cyclic generator a has prime order. The authors explicitly state that the general problem of finding all rotary pairs in S_n for n>5 remains open; the result in Lemma 4.1 resolves only the prime-order case.

References

It is an open question to find all rotary pairs $(a,z)$ for $_n$ with $n 5$; see Problem $1$ in .

— Regular, and (bi-)rotary Hall Cayley maps  (2608.30196 - Di et al., 31 Aug 2026) in Remark following Lemma 4.1 (Lemma~\ref{lem:RotofSp}), Section 4, subsection “Symmetric groups”