Computational Group Theory: Proceedings of the London by Michael D. Atkinson

By Michael D. Atkinson

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Suppose a TGQ S = T (O) is the point-line dual of a flock GQ S(F), F a Kantor semifield flock. Then T (O) is isomorphic to its translation dual T (O∗ ). We also have the following, which is due to J. A. Thas and H. Van Maldeghem [138]. 2 (J. A. Thas and H. Van Maldeghem [138]). Suppose that the TGQ T (O), with O = O(n, 2n, q) and q odd, is the point-line dual of a flock GQ S(F), where the point (∞) of S(F) corresponds to the line η of Type (b) of T (O). Then T (O) is good at the element η if and only if F is a Kantor semifield flock.

E. Payne [82]. Let q = 35 . 5. The Other Known Flock GQ’s of Order (q 2, q), q Odd 37 t ∈ GF(q), define a semifield flock of the quadratic cone with equation X0 X1 = X22 of PG(3, q). The flock, which is called the Penttila-Williams flock, was constructed by L. Bader, G. Lunardon and I. Pinneri in [4] using the Penttila-Williams ovoid of Q(4, 35 ) defined in [100]. The corresponding GQ, that is, the translation dual of S(F)D , is therefore referred to as the (sporadic) Penttila-Williams generalized quadrangle.

Proof. It is clear that if v and w are non-collinear points of p⊥ which are fixed by a whorl about p, then every point of the span {v, w}⊥⊥ is also fixed by the whorl. Now suppose Np is the net which arises from p, and suppose that Np is the (not necessarily proper) subnet of Np of order t + 1 which is generated by u, q and r. Then every point of Np is fixed by φ by the previous observation. 1 it is an affine plane of order t and s = t2 . Also, there arises a proper subquadrangle S of S of order t. 4 it follows that there is a proper subquadrangle Sφ of order (s , t), s = 1, which is fixed pointwise (and then also linewise) by φ.

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