Extended generalized quadrangles (roughly, connected structures whose every residue is a generalized quadrangle) are studied in some detail, especially those which are uniform or strongly uniform. Much basic structure theory is developed, many examples are given, and something approaching characterization is given for many types.

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Given a non-singular quadratic form q of maximal Witt index on V := V (2n + 1, F), let Δ be the building of type B n formed by the subspaces of V totally singular for q and, for 1k . In this paper we give a new very easy proof of this fact. We also prove that if char(F) = 2 then dim(As a consequence, when 1 < k < n and char(F) = 2 the embedding ε k is not universal. Finally, we prove that if F is a perfect field of characteristic p > 2 or a number field, n > k and k = 2 or 3, then ε k is universal.

Let $\Delta$ be a thick dual polar space of rank $n \geq 2$ admitting a full polarized embedding $e$ in a finite-dimensional projective space $\Sigma$, i.e., for every point $x$ of $\Delta$, $e$ maps the set of points of $\Delta$ at non-maximal distance from $x$ into a hyperplane $e^\ast(x)$ of $\Sigma$. Using a result of Kasikova and Shult , we are able the show that there exists up to isomorphisms a unique full polarized embedding of $\Delta$ of minimal dimension. We also show that $e^\ast$ realizes a full polarized embedding of $\Delta$ into a subspace of the dual of $\Sigma$, and that $e^\ast$ is isomorphic to the minimal full polarized embedding of $\Delta$. In the final section, we will determine the minimal full polarized embeddings of the finite dual polar spaces $DQ(2n,q)$, $DQ^-(2n+1,q)$, $DH(2n-1,q^2)$ and $DW(2n-1,q)$ ($q$\ud odd), but the latter only for $n\leq 5$. We shall prove that the minimal full polarized embeddings of $DQ(2n,q)$, $DQ^-(2n+1,q)$ and $DH(2n-1,q^2)$ are the `natural' ones, whereas this is not always the case for $DW(2n-1,q)$

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