Flocks of quadratic cones, generalized quadrangles and translation planes

The classification of spreads in PG(3,q) admitting linear groups of order q(q + 1), II. Even order

“…If the group $ is isomorphic to Sz(2 6 ), there is a L neburg-Tits subplane of order 2 2b by Theorem 23. However, 2 b ^ q/2, implying that 2 20 ^ q 2 /4.…”

“…Let h = p l , where p is a prime. Then there is a prime divisor u of h -1 but not of p s -1 for s < t unless t = 2 and p -f l = 2 a or p 1 = 2 6 .…”

“…By the work of Gevaert and Johnson [6], and Gevaert, Johnson and Thas [7], it is also true that within the collineation group of the translation plane associated with a so-called 'conical flock' (flock of a quadratic cone) is an elation group E of order q. In general, we shall use the term 'regulus-inducing' to denote such an elation group.…”

Classification of flocks of the quadratic cone over fields of order at most 29

“…Two partial BLT-sets are equivalent if they are in the same orbit of the automorphism group ΡΓΟ(5,#). A partial BLT-set of 0(4, q) has size at most q+ 1; if equality occurs it is a BLT-set: The known infinite families of BLT-sets are the classical BLT-sets associated with the linear flocks, the Fisher-Thas-Walker BLT-sets [15], [43] for fields of order congruent to 2 modulo 3, the Fisher BLT-sets [15] (see also [34]), the Kantor semifield BLT-sets [22] for field of non-prime order, the Kantor monomial BLT-sets [22] for fields of order congruent to 2 or 3 modulo 5, the Ganley BLT-sets [17], [16] for fields of characteristic 3, the Kantor likeable BLT-sets [17], [21] for fields of characteristic 5, the Mondello BLT-sets [34] for fields of order congruent to 1 or 4 modulo 5, and the Law -Penttila BLT-sets [25] for fields of characteristic 3. When there is a unique flock arising from the BLT-set, we give it the same name, except for the linear flocks arising from the classical BLT-sets (this covers the Fisher-Thas-Walker, Fisher, Kantor semifield and Mondello cases [3], [33], [34]).…”

“…(Recently Durante-Siciliano [14] gave a beautiful and short new proof.) That the Laguerre case was considerably more complicated was evident by then from constructions in [15], [43], [20], [30], [22], [41], [17], [16], [21], [3], [19], [31], [18]. However, in compensation, the link with generalised quadrangles [41], [23] makes these flocks more interesting.…”