Alice Devillers, Cai Heng Li, Geoff Pearce, Cheryl Praeger and I have just uploaded to the arxiv a preprint of our recently submitted paper `On imprimitive rank 3 permutation groups‘.
A permutation group on a set also acts on the set via . If is transitive on then we cannot expect it to be transitive on as the subset is an orbit. The best that we can hope for is that has two orbits on and this is equivalent to acting 2-transitively on , that is, acts transitively on the set of ordered pairs of distinct elements of . The rank of a permutation group is the number of orbits of on . Thus 2-transitive groups have rank 2.
The number of orbits of on is equal to the number of orbits of the point stabiliser on . Indeed, given an orbit of on the set is an orbit of on . Conversely, given an orbit of on , the set is an orbit of on . In this set up, corresponds to .
The study of rank 3 permutation groups goes back to Donald G. Higman in his paper `Finite permutation groups of rank 3′, Math. Z. 86, 145–156 (1964). A rank 3 group may be primitive or imprimitive. If it is primitive there are three possibilities:
- is an almost simple group,
- is a subgroup of the affine group for some prime
- acting on in product action where is a 2-transitive group on the set .
After the Classification of 2-transitive groups, work by Bannai in the alternating group case, Kantor and Liebler in the classical group case, Liebeck in the affine group case, and Liebeck and Saxl in the remaining cases of the sporadics and exceptional groups, all primitive rank 3 groups are known.
If is imprimitive on with a system of imprimitivity with parts, then must be 2-transitive on and for , is 2-transitive on . Moreover, Higman showed that is the unique system of imprimitivity for . If we let be the permutation group induced by on and is the group induced by on , then using the “Embedding Theorem” for imprimitive groups we have that is a subgroup of in its imprimitive action.
All 2-transitive groups are known and by a classical theorem of Burnside are either almost simple or affine, that is, the unique minimal normal subgroup of is either nonabelian simple or elementary abelian. We can think of the rank 3 group to be “large” if it contains while it is “small” if , that is, if acts faithfully on . The set of orbits of a normal subgroup of forms a system of imprimitivity. Since is the unique system of imprimitivity for the latter case implies that all normal subgroups of are transitive on , that is, is quasiprimitive on .
Our first main theorem is that if is an imprimitive group with the above set up such that is an almost simple 2-transitive group then is rank 3 if and only if one of the following holds.
- , that is, is “large”,
- is quasiprimitive and rank 3 on ,
- acting on 12 points,
- acting on 24 points.
We then classify all the quasiprimitive groups of rank 3 which are imprimitive. We show that they must all be almost simple groups and we find 8 individual groups plus certain actions of groups containing acting with blocks. Combined with the previous classification of all primitive rank 3 groups this implies that all quasiprimitive rank 3 groups are now known. The open case in the classification of all rank 3 groups is then the imprimitive case where the group induced on a block is a 2-transitive affine group. We say a little bit about this case in the paper.
We achieved our classification of quasiprimitive rank 3 groups by determining all almost simple groups which have a system of imprimitivity such that acts 2-transitively on and for we have is 2-transitive on . As mentioned earlier this includes all rank 3 imprimitive groups which are almost simple.
We became interested in a classification of these groups during our work on locally 2-distance transitive graphs. These are graphs where the stabiliser of a vertex acts transitively on the set of vertices at distance 1 from and on the set of vertices at distance 2 from . If the graph is also vertex-transitive then we refer to it as being 2-distance transitive. During our analysis, one case which arose was that of complete multipartite graphs with a group of automorphisms which acts faithfully on the partition into partite blocks. The group will be then be 2-distance transitive if it has rank 3. Our classification of quasiprimitive rank 3 groups which are imprimitive then gives a complete list of all possibilities which arise.