Abstract
All known finite generalized quadrangles that admit an automorphism group acting sharply transitively on their point set arise by Payne derivation from thick elation generalized quadrangles of order s with a regular point. In these examples only two groups occur: elementary abelian groups of even order and odd order Heisenberg groups of dimension 3. In [2] the authors determined all generalized quadrangles admitting an abelian group with a sharply transitive point action. Here, we classify thick finite generalized quadrangles admitting an odd order Heisenberg group of dimension 3 acting sharply transitively on the points. In fact our more general result comes close to a complete solution of classifying odd order Singer p-groups.
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De Winter, S., Thas, K. Generalized quadrangles admitting a sharply transitive Heisenberg group. Des. Codes Cryptogr. 47, 237–242 (2008). https://doi.org/10.1007/s10623-007-9146-6
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DOI: https://doi.org/10.1007/s10623-007-9146-6