The automorphism group of Payne derived generalized quadrangles. (English) Zbl 1129.51005
If \(Q\) is a generalized quadrangle of order \(s \geq 2\) having a regular point, then by a construction of Payne, a new generalized quadrangle \(P(Q,x)\) of order \((s-1,s+1)\) can be obtained. In the paper under review the authors prove that if \(Q\) is a generalized quadrangle \(Q\) of order \(s \geq 5\), \(s\) odd, with a center of symmetry \(x\), then every automorphism of \(P(Q,x)\) is induced by an automorphism of \(Q\) fixing \(x\).
Reviewer: Bart De Bruyn (Gent)
MSC:
| 51E12 | Generalized quadrangles and generalized polygons in finite geometry |
References:
| [1] | Ahrens, R. W.; Szekeres, G., On a combinatorial generalization of 27 lines associated with a cubic surface, J. Aust. Math. Soc., 10, 485-492 (1969) · Zbl 0183.52203 |
| [2] | Benson, C. T., On the structure of generalized quadrangles, J. Algebra, 15, 443-454 (1970) · Zbl 0212.52304 |
| [3] | Grundhöfer, T.; Joswig, M.; Stroppel, M., Slanted symplectic quadrangles, Geom. Dedicata, 49, 143-154 (1994) · Zbl 0801.51006 |
| [4] | Hall, M., Affine Generalized Quadrilaterals, Studies Pure Math. (1971), Academia Press: Academia Press London, pp. 113-116 · Zbl 0225.05021 |
| [5] | Hering, C.; Kantor, W. M.; Seitz, G. M., Finite groups with a split BN-pair of rank 1, I, J. Algebra, 20, 435-475 (1972) · Zbl 0244.20003 |
| [6] | Huppert, B., Endliche Gruppen I (1967), Springer-Verlag: Springer-Verlag Berlin · Zbl 0217.07201 |
| [7] | Miller, M. A.; Payne, S. E., Collineations and characterizations of generalized quadrangles of order \((q + 1, q - 1)\), Combinatorics ’98. Combinatorics ’98, Mondello. Combinatorics ’98. Combinatorics ’98, Mondello, Rend. Circ. Mat. Palermo (2) Suppl., 53, 137-166 (1998) · Zbl 0931.51005 |
| [8] | Moore, E. H., The subgroups of the generalized finite modular group, Decennial Publications of the University of Chicago, 9, 141-190 (1904) |
| [9] | Payne, S. E., Non-isomorphic generalized quadrangles, J. Algebra, 18, 201-212 (1971) · Zbl 0215.22301 |
| [10] | Payne, S. E., The generalized quadrangle with \((s, t) = (3, 5)\), Proc. Boca Raton 1990. Proc. Boca Raton 1990, Congr. Numer., 77, 5-29 (1990) · Zbl 0880.51002 |
| [11] | Payne, S. E.; Thas, J. A., Finite Generalized Quadrangles, Res. Notes Math., vol. 110 (1984), Pitman: Pitman Boston · Zbl 0551.05027 |
| [12] | Shult, E. E., On a class of doubly transitive groups, Illinois J. Math., 16, 434-455 (1972) · Zbl 0241.20004 |
| [13] | Thas, J. A.; Thas, K.; Van Maldeghem, H., Translation Generalized Quadrangles, Ser. Pure Math., vol. 26 (2006), World Scientific: World Scientific Singapore · Zbl 1126.51007 |
| [14] | A. Wiman, Bestimmung aller Untergruppen einer doppelt unendlichen Reihe von einfachen Gruppen, Stockh. Akad. Bihang 25, part 1, No. 2, 47 S., 1899; A. Wiman, Bestimmung aller Untergruppen einer doppelt unendlichen Reihe von einfachen Gruppen, Stockh. Akad. Bihang 25, part 1, No. 2, 47 S., 1899 · JFM 31.0147.01 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
