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Reconstruction of classical geometries from their automorphism group
Author(s) -
Barbina S.
Publication year - 2007
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/jdl016
Subject(s) - mathematics , symplectic geometry , automorphism , affine transformation , group (periodic table) , pure mathematics , vector space , unitary state , categorical variable , space (punctuation) , automorphism group , affine group , countable set , field (mathematics) , combinatorics , physics , computer science , statistics , quantum mechanics , political science , law , operating system
Abstract Let V be a countably infinite‐dimensional vector space over a finite field F . Then V is ω‐categorical, and so are the projective space PG ( V ) and the projective symplectic, unitary and orthogonal spaces on V . Using a reconstruction method developed by Rubin, we prove the following result: let ℳ be one of the above spaces, and let be an ω‐categorical structure such that Aut (ℳ) ≅ Aut () as abstract groups. Then ℳ and are bi‐interpretable. We also give a reconstruction result for the affine group AGL ( V ) acting on V by proving that V as an affine space is interpretable in AGL ( V ).