Regular cycles of elements in finite primitive permutation groups

Cheryl Praeger, Pablo Spiga and I have just uploaded to the arxiv our preprint Finite primitive permutation groups and regular cycles of their elements. Everyone recalls from their first course in group theory that if you write a permutation in cycle notation, for example (1,2)(3,4,5)(6,7,8,9,10), then its order is the lowest common multiple of it cycle lengths. As I have just demonstrated, not every permutation must have a cycle whose length is the same as the order of the element. We call a cycle of an element g whose length is the order of g, a regular cycle. A natural question to ask is when can you guarantee that a permutation has a regular cycle? Or in particular, for which permutation groups do all elements have a regular cycle?

Clearly, the full symmetric group contains elements with no regular cycles, but what about other groups? Siemons and Zalesskii showed that for any group G between PSL(n,q) and PGL(n,q) other than for (n,q)=(2,2) or (2,3), then in any action of G, every element of G has a regular cycle, except G=PSL(4,2) acting on 8 points. The exceptions are due to isomorphisms with the symmetric or alternating groups. They also later showed that for any finite simple group G,other than the alternating group, that admits a 2-transitive action, in any action of G every element has a regular cycle. This was later extended by Emmett and Zalesskii to any finite simple classical group not isomorphic to PSL(n,q).

With these results in mind, we started investigating elements in primitive permutation groups. The results of this work are in the preprint. First we prove that for $k\leq m/2$ , every element of Sym(m) in its action on k-sets has a regular cycle if and only if m is less than the sum of the first k+1 primes. After further computation we were then willing to make the following conjecture:

Conjecture: Let $G\leqslant Sym(\Omega)$ be primitive such that some element has no regular cycle. Then there exist integers $k\geq1, r\geq1$ and $m\geq5$ such that
G preserves a product structure on $\Omega=\Delta^r$ with $|\Delta|=\binom{m}{k}$ , and $Alt(m)^r \vartriangleleft G\leqslant Sym(m)\textrm{wr} Sym(r)$ , where Sym(m) induces its k-set action on $\Delta$ .

The general philosophy behind why such a result should be true is that most primitive groups are known to be small in terms of a function of the degree. There is a large body of work bounding the orders of primitive permutation groups with the best results due to Maroti. The actions of Sym(m) on k-sets are the usual exceptions.

Our paper goes about attempting to prove this conjecture. We make substantial progress and reduced its proof to having to deal with all the primitive actions of classical groups. Note that Emmett and Zalesskii only dealt with simple ones. One consequence of our work is we showed that every automorphism of a finite simple group has a regular cycle in its action on the simple group.

Pablo and Simon Guest have subsequently gone on to prove the result for all actions of classical groups and so the conjecture is now a theorem.

Pablo gave a great talk about the conjecture and its subsequent proof at the recent BIRS workshop on Permutation Groups in Banff which you can view here.