Point regular automorphism groups of generalised quadrangles

John and I have just uploaded to the arxiv a copy of our recent paper, `Point regular automorphism groups of generalised quadrangles‘. We investigate the regular subgroups of some of the known generalised quadrangles. We demonstrate that the class of groups which can act as a point regular group of automorphisms of a generalised quadrangle is much wilder than previously thought.

A permutation group {G} on a set {\Omega} acts regularly on a set {\Omega} if it acts transitively on {\Omega} and only the identity of {G} fixes an element of {\Omega}. Studying regular automorphism groups of projective planes has received much attention over the years. Recently attention has turned to the study of groups acting regularly on generalised quadrangles.

Dina Ghinelli proved in 1992 that a Frobenius group or a group with a nontrivial centre cannot act regularly on the points of a generalised quadrangle of order {(s,s)}, where {s} is even. Stefaan De Winter and Koen Thas proved in 2006 that if a finite thick generalised quadrangle admits an abelian group of automorphisms acting regularly on its points, then it is the Payne derivation of a translation generalised quadrangle of even order. Satoshi Yoshiara proved that there are no generalised quadrangles of order {(s^2 , s)} admitting an automorphism group acting regularly on points.

Our first result is a complete classification of all regular subgroups of the thick classical generalised quadrangles.

Theorem 1 Let {\mathcal{Q}} be a finite thick classical generalised quadrangle and let {G} be a group of automorphisms that acts regularly on the points of {\mathcal{Q}}. Then one of the following holds:

  1. {\mathcal{Q}=\mathsf{Q}^-(5,2)} and {G} is an extraspecial group of order 27 and exponent 3.
  2. {\mathcal{Q}=\mathsf{Q}^-(5,2)} and {G} is an extraspecial group of order 27 and exponent 9.
  3. {\mathcal{Q}=\mathsf{Q}^-(5,8)} and {G\cong \mathrm{GU}(1,2^9).9\cong C_{513}\rtimes C_9}.

Our proof uses the classification of regular subgroups of almost simple groups due to Martin Liebeck, Cheryl Praeger and Jan Saxl. While their results rely on the Classification of Finite Simple Groups, the part that we require only really depends on knowledge of the subgroup structure of low dimensional classical groups which was known before the classification. An alternative approach was independently undertaken by De Winter, K Thas and Shult.

Our other main result looks at the generalised quadrangles of order {(q-1,q+1)} obtained by Payne derivation from {\mathcal{Q}=\mathsf{W}(3,q)}. I described these quadrangles in an earlier post. The generalised quadrangle {\mathcal{Q}} has a group {G} of automorphisms which fixes a point {x} and each line on {x}, and acts regularly on the points not collinear with {x}. We call {G} the elation group. Moreover, {G} is elementary abelian for {q} even and a Heisenberg group for {q} odd. (A Heisenberg group is a Sylow {p}-subgroup of {\mathrm{GL}(3,q)}, that is, isomorphic to the group of lower triangular matrices with all diagonal entries equal to 1.)

The stabiliser {H} of the point {x} in the full automorphism group of {\mathcal{Q}} acts as a group of automorphisms of the Payne derived quadrangle {\mathcal{Q}^x} and contains {G} as a normal subgroup. However, this group may contain point regular subgroups other than {G}. In our paper we exploit this fact to construct several other infinite families of regular subgroups. Our results are summarised in the following theorem.

Theorem 2 Let {\mathcal{Q}^x} be the generalised quadrangle of order {(q-1,q+1)} obtained by Payne derivation from {\mathsf{W}(3,q)}. Then there exist subgroups {E,P} of {\mathrm{Aut}(\mathcal{Q}^x)} that act regularly on the points of {\mathcal{Q}^x} and for {q} not a prime there also exists a further regular subgroup {S} such that {E}, {P} and {S} have the following properties:

  1. {E} is an elation group of {\mathsf{W}(3,q)} while {P} and {S} are not;
  2. {E\not\cong S} and for {q} a power of {2} or {3}, {E\not\cong P};
  3. {S} is not special;
  4. For {q} even, {E} is elementary abelian while {S} and {P} have exponent 4 and are nonabelian except when {q=2};
  5. For {q=3^f}, {E} has exponent {3} while {P} and {S} have exponent {9};
  6. For {q} odd, {Z(P)=P'=Z(E)=E'} and {Z(S)<S'};
  7. For {q} even, {P'<Z(P)} and {S'<Z(S)}.

Explicit details and constructions can be found in the paper. The only groups previously known to act regularly on a GQ were elementary abelian 2-groups and Heisenberg groups of odd order. The groups we obtain include nonabelian 2-groups and {p}-groups which are not special (A {p}-group {P} is called special if the center of {P} is equal to the derived subgroup of {P} and {P/Z(P)} is elementary abelian).

The generalised quadrangle of order {(2,4)} obtained by Payne derivation from {\mathsf{W}(3,3)} is isomorphic to {\mathsf{Q}^-(5,2)}. The regular groups {E} and {P} occurring in our second theorem for this case are the two regular subgroups which appeared in our classification of regular subgroups of classical GQs.

Our paper includes computer calculations enumerating all regular subgroups of small GQs, and these demonstrate that there are many possible groups which act regularly on GQs, including {2}-groups with nilpotency class 7.

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