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

We tell the history of Moufang generalized quadrangles. We review some Moufang-like conditions, and classify finite quasi-transitive generalized quadrangles. These are generalized quadrangles so that for any two non-concurrent lines image of the GQ, and for some image, the group of generalized homologies with axes U and V acts transitively on the points of W 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 (!).