Quotients of geometries

Last week I uploaded to the arxiv the latest in a sequence of papers that introduces a “normal quotient” method for the study of incidence geometries. The first paper was `Quotients of incidence geometries‘ with Philippe Cara, Alice Devillers and Cheryl Praeger, the second was `Basic and degenerate geometries‘ with Cai Heng Li, Geoffrey Pearce and Cheryl Praeger, and the third was `Basic coset geometries‘ with Geoffrey and Cheryl. I will give a brief outline of the whole project here including motivation and some of the main results.

The study of graphs via their quotients has proved a very fruitful avenue of investigation. The general method for studying a particular class of graphs is to identify some reduction process that takes any graph in the family to a “basic graph” in the family, determine all “basic graphs”, and then determine all ways of lifting the “basic graphs” so that we can enumerate all graphs in the class. Examples of classes of graphs where this has proved a useful method of attack are:

  • distance-transitive graphs: Distance-transitive graphs with an imprimitive group of automorphisms are either bipartite or antipodal and can be reduced to smaller distance-transitive graph. Hence the basic graphs are the ones with a primitive group of automorphisms.
  • {s}-arc transitive graphs: If {G} acts transitively on the set of {s}-arcs, for {s\geq 2} of a graph {\Gamma} and {G} has a normal subgroup {N} with at least 3 orbits on vertices then the quotient graph {\Gamma_N} is also {s}-arc transitive and {\Gamma} is cover of {\Gamma_N}. Hence the basic graphs to study are those for which every nontrivial normal subgroup of {\mathrm{Aut}(\Gamma)} has at most two orbits, that is, {\mathrm{Aut}(\Gamma)} is quasiprimitive (all nontrivial normal subgroups are transitive) or biquasiprimitive (all have at most 2 orbits but not all are transitive).
  • vertex-intransitive, locally {s}-arc transitive graphs: Here the graphs are bipartite and the bipartite halves are the two orbits of {G} on vertices. We can apply the same quotienting process and the basic graphs are those for which {G} is quasiprimitive on at least one orbit.

In all cases we can use the O’Nan-Scott Theorem for primitive or quasiprimitive groups to study the basic graphs. This involves seeing which primitive/quasiprimitive types are possible and then determining all graphs for a given type. Usually we can reduce to studying the case where the action is either almost simple or affine.

The aim of our project on incidence geometries is to extend these methods to geometries. Jacques Tits introduced the notion of a building to study the simple groups of Lie type. Francis Buekenhout then introduced incidence geometries and their associated diagrams as a generalisation in an attempt to find a larger class of geometries that would include buildings but also contain geometries associated to the sporadic simple groups. One rationale for our project was to give a geometrical justification for the study of geometries of simple groups. The hope was that geometries of almost simple groups would arise as one of a small number of “basic” families of graphs.

First some definitions. A pregeometry {\Gamma = (X,\ast,t)} consists of a set {X} of elements (often called points) with an incidence relation {\ast} on the points, and a map {t} from {X} onto a set {I} of types. The incidence relation is symmetric and reflexive, and if {x \ast y} we say that {x} and {y} are incident. Furthermore if {x \ast y} with {x \neq y} then {t(x) \neq t(y)}. The number {|I|} of types is called the rank of the pregeometry. A rank 2 geometry is merely a bipartite graph and so our earlier work on quotients of locally {s}-arc transitive graphs can be seen as investigating the rank 2 case.

A flag {F} is a set of pairwise incident elements of {\Gamma} (which implies that the elements of {F} are of pairwise distinct types). The type of a flag is the set of types of its elements. A chamber is a flag containing one element of each type. A pregeometry in which every flag is contained in a chamber is called a geometry. The typical example is a projective space. Here the elements are the points, lines, planes etc and incidence is given by inclusion.

Let {\Gamma=(X,*,t)} be a pregeometry and let {\mathcal{B}} be a partition of {X} which is a refinement of the type partition of {\Gamma}. We define a new pregeometry {\Gamma_{/\mathcal{B}}=(\mathcal{B},*_{/\mathcal{B}},t_{/\mathcal{B}})} where

  1. {B_1 *_{/\mathcal{B}} B_2} if and only if there exist {\alpha_i\in B_i} (for {i=1,2}) with {\alpha_1*\alpha_2},
  2. {t_{/\mathcal{B}}:X_{\mathcal{B}}\rightarrow I} and {t_{/\mathcal{B}}(B)=t(\alpha)} for {\alpha\in B}.

Quotients of geometries under certain conditions were studied by Tits and Pasini. However, in general the quotient of a geometry is not necessarily a geometry. Our first paper investigated necessary and sufficient conditions for when the quotient of a geometry is a geometry, and also which geometrical properties are preserved by quotients.

The automorphism group of a pregeometry is the group of all permutations of the element set {X} that preserves the type of each element and that preserves incidence. We say that a group of automorphisms {G} is vertex-transitive on the pregeometry if {G} is transitive on each set {X_i=t^{-1}(i)} for {i\in I}, that is, is transitive on each set of points of a given type. That is, the orbits of {G} on points are as large as possible. We say that {G} is incidence-transitive if for each pair {\{i,j\}} of types, {G} is transitive on the set of flags of type {\{i,j\}}. We say that {G} is flag-transitive if for each {J\subseteq I}, {G} is transitive on the set of flags of type {J}.

One class of pregeometries are coset pregeometries. Given a group {G} with subgroups {\{G_i\}_{i\in I}}, the coset pregeometry {\Gamma=\Gamma(G,\{G_i\}_{i\in I})} is the pregeometry whose elements of type {i\in I} are the right cosets of {G_i} in {G} and two cosets {G_ix} and {G_jy} are incident if {i\neq j} and {G_ix\cap G_jy} is nonempty. The group {G} acts by right multiplication on the elements of {\Gamma} inducing automorphisms of the pregeometry, that is, maps incident elements to incident elements. In our first paper we showed that a pregeometry is a coset pregeometry if and only if its group of automorphisms is incidence-transitive and vertex-transitive and the pregeometry contains a chamber.

One of the main results of our first paper is that the quotient of a coset pregeometry is still a coset pregeometry. This motivated us to take the class of coset pregeometries with connected rank 2 truncations as the class of objects that we should study, instead of the narrower class of flag-transitive geometries.

The next thing to do is to determine what the degenerate and basic coset pregeometries should be. The basic pregeometries are those such that all quotients are degenerate. We investigated this question in our second paper. The definitions will depend on whether we are taking normal quotients (that is quotients with respect to the set of orbits of some normal subgroup) or imprimitive quotients, (that is quotients with respect to some {G}-invariant partition). In the normal quotients case we were able to show that the basic pregeometries can be built from pregeometries where for each type {i}, the action of {G} is quasiprimitive on each set of elements of type {i}. In the imprimitive quotients case we replace `quasiprimitive’ with `primitive’.

Thus the important pregeometries to study are those where {G} is quasiprimitive on each set of elements of type {i}. We investigated these in our third paper which contains many constructions of flag-transitive geometries. One type of primitive action is the almost simple case. The projective space {PG(n,q)} is an example of an incidence geometry of rank {n} and the automorphism group {P\Gamma L(n+1,q)} is primitive of almost simple type on each set of elements. Our initial aim was to see if there was some number {N} such that if {\Gamma} were a flag-transitive geometry of rank at least {N} such that the automorphism group {G} acts primitively on each set of elements then {G} is almost simple. However, the main result of our third paper was to show that this is not true. In fact for each positive integer {n} and each of the 8 O’Nan-Scott types of primitive group, we were able to find a primitive group {G} on a set {\Omega} whose action is primitive of that type and construct a rank {n} geometry {\Gamma} such that each set of elements of a given type is a copy of {\Omega} and {G} acts on flag-transitively on {\Gamma}.


The O’Nan-Scott Theorem

I have decided that any mathematical blog entitled SymOmega should have an expository post about one of the most influential theorems of permutation group theory, that is, The O’Nan-Scott Theorem. Originally, this was a theorem about maximal subgroups of the symmetric group.  It appeared as an appendix to a paper of Leonard Scott in the proceedings of the Santa Cruz symposium on finite simple groups in 1979, with a footnote that Michael O’Nan had independently proved the same result. Apparently there are even earlier versions due to various people but it was the Classification of Finite Simple Groups (CFSG) which meant that it would become very useful.

In its simplest form the theorem states that a maximal subgroup of the symmetric group \mathrm{Sym}(\Omega) where |\Omega|=n, is one of the following:

  1. S_k\times S_{n-k}, the stabiliser of a k-set (that is, intransitive),
  2. S_a \mathrm{wr} S_b with n=ab, the stabiliser of a partition into b parts of size a (that is, imprimitive), or
  3. primitive (that is, preserves no nontrivial partition) and of one of the following types:
  • \mathrm{AGL}(d,p),
  • S_\ell \mathrm{wr} S_k with n=\ell^k, the stabiliser of a product structure \Omega=\Delta^k,
  • a group of diagonal type, or
  • an almost simple group.

(These  types will be explained in further detail below).

It was soon recognised (perhaps first in a paper of Peter Cameron)  that the real power is in the ability to split the finite primitive groups into various types.  Problems concerning primitive groups can then be studied by solving them for each type.  This often sees questions about primitive groups reduced to questions about simple groups and then the force of the classification can be harnessed to get your result.  Furthermore, many results about transitive permutation groups can be reduced to the primitive case and so we actually have a tool for studying questions about transitive groups.

The division of primitive groups into various types is usually finer than given in the statement about maximal subgroups of S_n as we are no longer concerned about maximality in S_n but instead are more concerned about the types of actions. In Cameron’s original paper, the twisted wreath product case (again, more details later) which does not appear as a maximal subgroup of S_n but is an important type of action with a distinctly different flavour to the maximal subgroup S_\ell \mathrm{wr} S_k  it is contained in,  was left out and this was corrected in a subsequent paper of Aschbacher and Scott, and in work of Laci Kovacs. A complete self-contained proof is given in a paper by Martin Liebeck, Cheryl Praeger and Jan Saxl.

Since I am from Perth, I usually  follow the  division and labelling into 8 types due to Cheryl Praeger which appears here and here.  First we need to deal with a few preliminaries.

Continue reading “The O’Nan-Scott Theorem”

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