Generalised quadrangles V: elation generalised quadrangles

The focus of this post is elation generalised quadrangles. These are generalised quadrangles defined with respect to certain automorphisms of the generalised quadrangle. My main sources for this post have been Michel (Celle) Lavrauw’s and Maska Law’s Phd theses which are both availiable from Ghent’s PhD theses in finite geometry page.

Let {\mathcal{Q}} be a generalised quadrangle of order {(s,t)} with a point {x}. An elation about {x} is an automorphism of {\mathcal{Q}} that fixes {x}, fixes each line incident with {x} and fixes no point not collinear with {x}. If there exists a group {G} of order {s^2t} consisting entirely of elations and which acts regularly on the set of points not collinear with {x} then {\mathcal{Q}} is called an elation generalised quadrangle or simply an EGQ. The group {G} is called an elation group and {x} is called the base point.

Many of the known examples of generalised quadrangle are EGQs. For example, consider {\mathsf{W}(3,q)} defined with respect to the alternating form {B(u,v)=uJv^T} where

\displaystyle J=\begin{pmatrix} 0&0&0&1\\ 0&0&1&0\\ 0&-1&0&0\\ -1&0&0&0 \end{pmatrix}

Let {x} be the point given by the 1-dimensional subspace spanned by {(1,0,0,0)}. Now {\mathrm{Sp}(4,q)} contains the subgroup

\displaystyle E=\left\{ \begin{pmatrix} 1&0&0&0\\ -c&1&0&0\\ b&0&1&0\\ a&b&c&1\end{pmatrix}\big\vert a,b,c\in\mathrm{GF}(q)\right\}

Then {x^\perp=\langle (1,0,0,0),(0,1,0,0),(0,0,1,0)\rangle} and {E} acts trivially on {x^{\perp}/x}, so {E} fixes each line incident with {x}. Moreover, the points not collinear with {x} are all points of the form {\langle (x,y,z,1)\rangle} such that {x,y,z\in\mathrm{GF}(q)}. The group {E} acts regularly on this set and so each element of {E} is an elation and {\mathsf{W}(3,q)} is an EGQ. When {q} is even, {E} is abelian while when {q} is odd {E} is known as a Heisenberg group. In particular, for {q} a prime {E} is an extraspecial group of exponent {p}.

Of all the examples we have seen so far the only generalised quadrangles which are not EGQ’s and nor are their duals, are the Payne-derived quadrangles of order {(q-1,q+1)}.

We should note here a couple of points:

  • Not every point of an EGQ need be a base point. Indeed it was shown by Koen Thas and Hendrik van Maldeghem in 2008 that if every point of an EGQ is a base point then the EGQ is either a classical GQ or the dual of a classical GQ.
  • It was recently noticed by Stan Payne and Koen Thas that the set of all elations about of an EGQ does not necessarily form a group.
  • It was proved by Robert Rostermundt that {\mathsf{H}(3,q^2)} for {q} even has two nonisomorphic elation groups.
  • It is not known if the elation group of a EGQ need be a {p}-group but it is conjectured to be so.

Kantor families

In 1980 Bill Kantor discovered a general way of constructing EGQs as coset geometries as follows:

Let {G} be a group of order {s^2t} with {s,t>1}. Let {\{A_i\}_{i=0}^t} be a family of subgroups of {G} of order {s} and {\{A_i^*\}_{i=0}^t} be a family of subgroups of order {st} such that for each {i}, {A_i\leqslant A_i^*}. We then define an incidence structure with points

  • (i) the elements of {G},
  • (ii) the right cosets {A_i^*g},
  • (iii) a point {(\infty)};

and lines

  • (a) the right cosets {A_ig},
  • (b) the symbols {[A_i]}.

Incidence is defined as follows: a point {g\in G} of type (i) is incident with each line {A_ig}, a point {A_i^*g} of type (ii) is incident with the line {[A_i]} and each line {A_ih} contained in {A_i^*g}, and the point {(\infty)} is incident with each line {[A_i]}.

The incidence structure just defined is a GQ of order {(s,t)} if and only if

  1. {A_iA_j\cap A_k=1} for distinct {i,j,k},
  2. {A_i^*\cap A_j=1} for {i\neq j}.

A collection of such subgroups {\{A_i\}_{i=0}^t}, {\{A_i^*\}_{i=0}^t} in a group {G} yielding a GQ is called a (Kantor) 4-gonal family.

The group {G} acts as a group of automorphisms of our GQ by fixing the symbols {[A_i]} and {(\infty)}, and acting on the remaining points and lines by right multiplication. In particular, each element of {G} is an elation about {(\infty)} and so we have constructed an EGQ.

Not only does a Kantor 4-gonal family give rise to an EGQ, but each EGQ can be constructed in such a manner. We now outline this construction.

Let {\mathcal{Q}} be an EGQ with elation group {G} and base point {x}. Since {G} acts regularly on the set of points not collinear with {x}, this set of points can be identified with the elements of {G}. Let {\ell} be a line incident with {x}. By the GQ property, there is a unique line {\ell'} incident with the point {z'=1_G} and meeting {\ell}. Let {y'} be the point of intersection of {\ell} and {\ell'}. Now let {\ell''} be another line not incident with {x} but meeting {\ell} and let {y''} be the point of intersection. Let {z''=g\in G} be a point not collinear with {x} and incident with {\ell'}. Then {(z')^g=z''} and since {\mathcal{Q}} is an EGQ, {g} fixes {\ell}. By the GQ property, {\ell'} is the unique line incident with {z'} that meets {\ell} and {\ell''} is the unique line incident with {z''} that meets {\ell}. Thus {(\ell')^g=\ell''} and {(y')^g=y''}. In particular, {G} acts transitively on the set of lines not incident with {x} that meet {\ell}. There are {st} such lines, so {|G_{\ell'}|=s} and the set of such lines can be identified with the cosets of {G_{\ell'}} in {G}. We also have that {G} acts transitively on the set of points other than {x} that are incident with {\ell}. There are {s} such points and so {|G_{y'}|=st} and the set of such points can be indentified with the cosets of {G_{y'}} in {G}. Now {G_{ell'}\leqslant G_{y'}} and {G_{\ell'}g} is identified with the line {\ell''}, which is incident with {g}. Moreover, a point identified with {G_{y'}h}, for {h\in G} will be incident with the line {G_{\ell'}h} and note that {G_{\ell'}h} is contained in {G_{y'}h}. Running over all lines {\ell_i} incident with {x} we obtain {t+1} subgroup {\{G_{\ell'_i}\}} of order {s} and {t+1} subgroups {\{G_{y'_i}\}} of order {st}, and all lines not incident with {x} have been identified with a coset of a subgroup of order {s} and each point collinear with {x} has been identified with a coset of a subgroup of order {st}. Finally, we identify {x} with the symbol {(\infty)} and each line {\ell_i} incident with {x} with the symbol {[G_{\ell_i'}]}.

It remains to check that the collection of subgroups we have obtained satisfy the two conditions that guarantee that Kantor’s construction yields a GQ. Let {g\in G_{\ell_i'}\cap G_{y_j'}} for {i\neq j}. Since {G_{\ell_j'}\leqslant G_{y_j'}}, it follows that the coset {G_{\ell_j'}g} is identified with a line incident with {y_j'}. Moreover, the point identified with {g} is incident with this line. Since {g\in G_{\ell_i'}}, we also have that the point identified with {g} is incident with {\ell_i'}. Hence {G_{\ell_j'}g} is a line through {y_j'} meeting {\ell_i'} at {g}. By the GQ property there is a unique line through {y_j'} meeting {\ell_i'} and so {G_{\ell_i'}\cap G_{y_j'}=1}.

Finally, consider {G_{\ell_i'}G_{\ell_j'}\cap G_{\ell_k'}}. If {g\in G_{\ell_j'}} then {G_{\ell_i'}g} is the line through the point {g} of {\ell_j'} that meets {\ell_i}. In particular, {G_{\ell_i'}G_{\ell_j'}} is the set of all points not collinear with {x} that lie on a line that meets both {\ell_i} and {\ell_j'}. Hence if {g\in {\ell_i'}G_{\ell_j'}\cap G_{\ell_k'}} then {g} is a point of {\ell_k'} that also lies on a line meeting both {\ell_i} and {\ell_j'}. Now from the way we constructing the Kantor family, the point {1_G} lies on {\ell_i',\ell_j'} and {\ell_k'}. Thus if {g\neq 1} then the point {g} also lies on the line {G_{\ell_i'}g} that meets {\ell_j'}. Hence we have two lines through {g} ({\ell_k'} and {G_{\ell_i'}g}) that meet {\ell_j'}, contradicting the GQ property. Hence {G_{\ell_i'}G_{\ell_j'}\cap G_{\ell_k'}=1}.

I mentioned in my last post on flock generalised quadrangles that given a {q}-clan it is possible to construct a generalised quadrangle of order {(q^2,q)} via a group coset construction. This is done by constructing a group {G} and using the {q}-clan to construct the two families {\{A_i\}} and {\{A_i^*\}} of {q+1} subgroups of {G}. I will not go through this here but it is outlined in both Maska’s and Celle’s theses.

The next post will be the last in the series and will be on translation generalised quadrangles. Unlike this post it will include some GQs not seen so far.

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