Moufang-like conditions for generalized quadrangles and classification of all finite quasi-transitive generalized quadrangles

We tell the history ofMoufang generalized quadrangles.We review someMoufang-like conditions, and classify finite quasi-transitive generalized quadrangles. These are generalized quadrangles so that for any two non-concurrent linesU,V of theGQ, and for someW ∈ {U,V }⊥, the group of generalized homologies with axes U and V acts transitively on the points ofW incident with neither U nor V. As a by-product of the proof, we will show, without the classification of finite simple groups, that a finite generalized quadrangle that admits a BN-pair of rank 2 is classical or dual classical if and only if it admits at least one nontrivial homology (!). © 2005 Published by Elsevier B.V. MSC: 05B25; 05E20; 20B25; 20B27; 20E42; 51E12