My old friend Leonard Soicher is visiting St Andrews this week, having made a better job of retiring than I did. Today he gently took me to task for using the expression “simple diagonal type” for one of the types in the O’Nan–Scott Theorem in a manner slightly different from the official one. But there is an interesting story here, which I will tell.
The most commonly used type division in this theorem (which describes primitive permutation groups in terms of their socle) is the one by Cheryl Praeger. You can find a clear exposition of it in a post by Michael Giudici on the SymOmega blog.
Before getting to the point at issue, some preliminaries.
I like, where possible, to define permutation properties in a fairly uniform way, using the formula “A permutation group has property X if it preserves no non-trivial A-structure”. Here a trivial structure is one which is invariant under the full symmetric group on the domain. So, for example:
- If A-structure is “subset”, the trivial subsets are the empty set and the entire domain, and so the corresponding X is “transitive”.
- If A-structure is “partition”, the trivial partitions are the partition into singletons and the partition with a single part, and so the corresponding X is “primitive”.
- If A-structure is “graph”, the trivial graphs are complete and null, and the corresponding X is “2-homogeneous” (or “2-set transitive”).
- If A-structure is “digraph”, the corresponding X-structure is “2-transitive”.
A couple more in a moment. But first let me point out the problem. If the size of the domain is 2, then the only partitions are trivial ones, and so the trivial group would qualify as “primitive”. You might like to stop and think whether you want that or not.
So here are a couple more. If A-structure is “Hamming graph”, then X is what I have called “basic”; if A-structure is “graph with clique number equal to chromatic number”, then X is “synchronizing”.
So back to group theory. Let S be a group. The holomorph of S is the semidirect product of S by Aut(S), its automorphism group. If S is the additive group of a vector space over the field of p elements, for p prime, its holomorph is the affine group on this vector space. In general, Hol(S) acts as a permutation group on S, where S acts by right multiplication and its automorphism group acts in the natural way.
The case where S has trivial centre is the one relevant here. In this case, the inner automorphism group of S is isomorphic to S. Now consider the following three actions of S on itself:
- by right multiplication: x → xs;
- by left multiplication by the inverse: x → s−1x (the inverse is necessary to get the action working correctly);
- by conjugation: x → s−1xs.
It is clear that a permutation group which contains two of these actions will also contain the third. So the semidirect product of S by its inner automorphism group is isomorphic to the direct product of two copies of S (one acting on the right, the other on the left). As a passing remark, the commutativity of these two actions is equivalent to the associativity of elements of S, as you will see if you write it out.
So in this case, we can define the holomorph to be the extension of S×S by the outer automorphism group Out(S) of automorphisms modulo inner automorphisms (we already have the inner automorphisms).
Now, more generally, the “full” diagonal group Diag(S,m) is generated as a group by the right multiplications, automorphisms of S (acting simultaneously on all coordinates) and permutations of the coordinates. Subgroups of such a group containing the socle Sm are sometimes just called “diagonal groups”. It was worked out a long time ago (I am not quite sure by whom) that, if S is a non-abelian simple group, then a diagonal group with socle Sm (for m > 1) is primitive if and only if the subgroup of Sym(m) permuting the coordinates is primitive. But, warning: this is primitive in the weaker sense of not preserving any non-trivial partition; so this subgroup must be either the trivial group of degree 2, or a group which is primitive in the more usual sense.
Note that the element which interchanges the two factors in the socle acts on S as the inversion map x → x−1. So the full diagonal group is obtained from the holomorph of S by adjoining the inversion map.
Now turn back to Michael Giudici’s account. He defines a group to be of type (HS) if it is contained in the holomorph of a simple group, and of type (SD) if it is as described in the preceding paragraph (with the small modification that the group permuting the factors in the socle is primitive in the usual sense).
So it appears that two primitive groups with identical socles acting in identical ways can belong to different types in this classification depending on whether or not the inversion belongs to the group. This rather conflicts with the notion that the O’Nan–Scott Theorem classifies primitive groups by their socles.
This seems less than satisfactory to me; but perhaps I have misunderstood Cheryl’s classification.