AS-configurations and skew-translation generalised quadrangles

Abstract

The only known skew-translation generalised quadrangles (STGQ) having order (q, q), with q even, are translation generalised quadrangles. Equivalently, the only known groups G of order q 3 , q even, admitting an Ahrens-Szekeres (AS-)configuration are elementary abelian. In this paper we prove results in the theory of STGQ giving (i) new structural information for a group G admitting an AS-configuration, (ii) a classification of the STGQ of order (8,8), and (iii) a classification of the STGQ of order (q, q) for odd q (using work of Ghinelli and Yoshiara).Crown

The point regular automorphism groups of the Payne derived quadrangle of $W(q)$

“…(P1) The set U = F q for Constructions 6.1 and 6.3, and |U| = q/p for Constructions 6.2 and 6. 4. This follows by a case by case check.…”

“…For instance, it is shown in [31] that finite thick generalized quadrangles of order (t 2 , t) does not admit a point regular automorphism group. By combining the results in [14] and [31], it was shown in [4] that any skew-translation generalized quadrangle of order (q, q), q odd, is isomorphic to the classical symplectic quadrangle W (q). Swartz [27] initiated the study of generalized quadrangles admitting an automorphism group that acts regularly on both points and lines.…”

“…Lemma 2. 4. Let q = p m be a prime power with m = p e l, and take g ∈ Aut(F q ) such that g(x) = x p l .…”

“…Therefore, the two constructions yield non-isomorphic groups in the even characteristic case. 4. The structure of a nonlinear point regular group G of Q P Throughout this section, we assume that G is a nonlinear point regular group of the quadrangle Q P as defined in Theorem 2.9 with associated functions T and θ.…”

Central aspects of skew translation quadrangles, 1

“…Suppose that S is an elementary abelian 2group, and let |S| = 2 n = t. We see S as an affine n-space Π over F 2 . Then we have a natural homomorphism (36) η : H → AGL n (2) defined by conjugation; its kernel is C H (S). It might be interesting to study the action of the involutions in H on the space S.…”