Titlebook: q-Clan Geometries in Characteristic 2; Ilaria Cardinali,Stanley E. Payne Book 2007 Birkh?user Basel 2007 Dimension.automorphism group.boun

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Other Good Stuff,eal and have a wide variety of connections with other geometric objects. For a general reference see J. A. Thas and S. E. Payne [TP94]. For . = 2e see especially [BOPPR1] and [BOPPR2]. In this section we give a very brief introduction to the material contained in these latter two papers.

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Book 2007sarguesian plane of order 2. It is also equivalent to a flock of a quadratic cone, and hence to a line-spread of 3-dimensional projective space and thus to a translation plane, and more. These geometric objects are tied together by the so-called Fundamental Theorem of q-Clan Geometry.?The book gives

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The Adelaide Oval Stabilizers, which acts on the points of this line as (0, .) ? (0, y2/δ, ., from which it follows that exactly three points on this line are fixed: the oval point (0, ., 1) and two others: (0, 1, 0) and (0, 0, 1). But the secant line through . and . passes through the point (0, 1, 0), implying that the nucleus must be (0, 0, 1).

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Book 2007 a?complete proof of this theorem, followed by a detailed study of the known examples. The collineation groups of the associated generalized quadrangles and the stabilizers of their associated ovals are worked out completely..

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The Payne q-Clans,98] show that up to the usual equivalence of .-clans, the three known examples are the only ones. Since the two non-classical families exist only for . odd, we assume throughout this chapter that . is odd. Then the three known families have the following appearance. There is some positive integer .