Between transitive and primitive

Many familiar properties of permutation groups form a hierarchy. A few years ago, with João Araújo and Ben Steinberg, I wrote a survey on properties related to synchronization of finite automota, lying between primitivity and 2-transitivity. Here is a new property near the bottom of the hierarchy.

The notion of primitivity was introduced by Galois, and extensively discussed in his Second Memoir, and has been of great interest to mathematicians ever since. However, a careful reading of the text by Peter Neumann revealed that Galois confused two different notions. A transitive permutation group G is primitive if the only G-invariant partitions are the trivial ones (the partition into singletons, and the partition with a single part). However, sometimes Galois used a different property, namely that any non-trivial normal subgroup is transitive. This is weaker, since the orbits of a normal subgroup of G form a G-invariant partition; and it is strictly weaker, as can be seen by taking a transitive but imprimitive action of a simple group (for example, the regular action on itself by right multiplication). This property is now called quasiprimitivity.

I had the following idea. Suppose that P and R are properties with R strictly stronger than P. Can we find a property Q which is independent of P (in the sense that neither implies the other) but such that P and Q together are equivalent to R? Investigation of Q might throw some light on the difference between P and R. There is no obvious natural property which does this in general, but when one exists, I decided to call it pre–R.

In the case of quasiprimitivity and primitivity, there is such a property. A permutation group G is pre-primitive (or PP for short) if every G-invariant partition is the orbit partition of a subgroup of G. If this holds, it is easy to see that we can take the subgroup to be normal. From this it follows that pre-primitivity and quasiprimitivity imply primitivity. The converse is also clear, and it is easy to find examples having one of the two properties but not the other.

With the help of Marina Anagnostopoulou-Merkouri and Enoch Suleiman, I have been investigating this property; our paper is now on the arXiv, at 2302.13703. There is quite a lot to say. Here are a couple of sample results.

• A regular permutation group G is pre-primitive if and only if G is a Dedekind group (that is, every subgroup is normal).
• The wreath product of two transitive groups in its imprimitive action is pre-primitive if and only if both factors are pre-primitive. (Things are not so straightforward for other products such as direct product or wreath product in the product action, though in these cases we have necessary conditions and sufficient conditions.)
• Let T be a finite group and m a positive integer. Suppose that, for any characteristic subgroup K of T, if L is the subgroup of K generated by (m+1)th powers and commutators, then every subgroup of K containing L is normal in T. Then the diagonal group D(T,m) is pre-primitive. (It may be that the converse is also true, but we have not been able to prove this.)

One of our most remarkable discoveries is that we happened to strike it lucky with our first attempt. We looked at several other pairs of properties in the hierarchy and found very little to say about them:

• Transitivity and quasiprimitivity: G is pre-QP if every normal subgroup of G is either transitive or trivial on each orbit. So what?
• k-homogeneity (transitivity on k-element sets) and k-transitivity (transitivity on k-tuples of distinct elements): This had already been done by Peter Neumann, who called a group generously (k−1)-transitive if the setwise stabiliser of any k-set induces the symmetric group on it.
• Primitivity and synchronization: G is synchronizing if, whenever f is a self-map of the domain which is not a permutation, the monoid generated by G and f contains an element of rank 1. Peter Neumann (again) found a convenient version for our game. A partition Π of the domain is called section-regular if there is a subset A such that, for all g∈G, the set Ag is a section for Π. Now G is synchronizing if and only if it has no non-trivial section-regular partition. So we can say that G is pre-synchronizing if every section-regular partition is G-invariant. This does what we want (so that a group is synchronizing if and only if it is primitive and pre-synchronizing); but unfortunately the following holds: the only imprimitive pre-synchronizing group is the Klein group acting regularly.

As noted, there is much more; take a look at the paper, or try your hand at this game with other pairs of properties. There is no reason to stick to permutation groups here! In the meantime we have found another property, even weaker than PP, which we are still investigating; I hope to report on it sometime soon.