Isomorphisms of groups related to flocks
Koen Thas
Department of Mathematics, Ghent University, Krijgslaan 281, S25, 9000 Ghent, Belgium
DOI: 10.1007/s10801-011-0326-0
Abstract
A truly fruitful way to construct finite generalized quadrangles is through the detection of Kantor families in the general 5-dimensional Heisenberg group \mathbb F q \mathbb{F}_{q}. All these examples are so-called “flock quadrangles”. Payne (Geom. Dedic. 32:93-118, 1989) constructed from the Ganley flock quadrangles the new Roman quadrangles, which appeared not to arise from flocks, but still via a Kantor family construction (in some group 
of the same order as 
). The fundamental question then arose as to whether 
(Payne in Geom. Dedic. 32:93-118, 1989). Eventually the question was solved in Havas et al. (Finite geometries, groups, and computation, pp. 95-102, de Gruyter, Berlin, 2006; Adv. Geom. 26:389-396, 2006). Payne's Roman construction appears to be a special case of a far more general one: each flock quadrangle for which the dual is a translation generalized quadrangle gives rise to another generalized quadrangle which is in general not isomorphic, and which also arises from a Kantor family. Denote the class of such flock quadrangles by 
.
of the same order as ). The fundamental question then arose as to whether (Payne in Geom. Dedic. 32:93-118, 1989). Eventually the question was solved in Havas et al. (Finite geometries, groups, and computation, pp. 95-102, de Gruyter, Berlin, 2006; Adv. Geom. 26:389-396, 2006). Payne's Roman construction appears to be a special case of a far more general one: each flock quadrangle for which the dual is a translation generalized quadrangle gives rise to another generalized quadrangle which is in general not isomorphic, and which also arises from a Kantor family. Denote the class of such flock quadrangles by .Pages: 111–121
Keywords: keywords flock quadrangle; elation quadrangle; automorphism group; Heisenberg group; characterization
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References
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4. Bloemen, I., Thas, J.A., Van Maldeghem, H.: Elation generalized quadrangles of order (p, t), p prime, are classical. Special issue on orthogonal arrays and affine designs, Part I. J. Stat. Plan. Inference 56, 49-55 (1996)
5. Brown, M.R.: Projective ovoids and generalized quadrangles. Adv. Geom. 7, 65-81 (2007)
6. Cardinali, I., Payne, S.E.: q-Clan Geometries in Characteristic
2. Frontiers in Mathematics. Birkhäuser, Basel (2007)
7. Frohardt, D.: Groups which produce generalized quadrangles. J. Comb. Theory, Ser. A 48, 139-145 (1988)
8. Gorenstein, D.: Finite Groups, second edn. Chelsea, New York (1980)
9. Hachenberger, D.: Groups admitting a Kantor family and a factorized normal subgroup. Des. Codes Cryptogr. 8, 135-143 (1996). Special issue dedicated to Hanfried Lenz
10. Havas, G., Leedham-Green, C.R., O'Brien, E.A., Slattery, M.C.: Computing with elation groups. In: Finite Geometries, Groups, and Computation, pp. 95-102. de Gruyter, Berlin (2006)
11. Havas, G., Leedham-Green, C.R., O'Brien, E.A., Slattery, M.C.: Certain Roman and flock generalized quadrangles have nonisomorphic elation groups. Adv. Geom. 26, 389-396 (2006)
12. Johnson, N.L.: Semifield flocks of quadratic cones. Simon Stevin 61, 313-326 (1987)
13. Kantor, W.M.: Generalized quadrangles associated with G2(q). J. Comb. Theory, Ser. A 29, 212-219 (1980)
14. Kantor, W.M.: Generalized quadrangles, flocks and BLT sets. J. Comb. Theory, Ser. A 58, 153-157 (1991)
15. Payne, S.E.: Generalized quadrangles as group coset geometries. Congr. Numer. 29, 717-734 (1980). Proceedings of the Eleventh Southeastern Conference on Combinatorics, Graph Theory and Computing (Florida Atlantic University, Boca Raton, Fla., 1980), vol. II
16. Payne, S.E.: A garden of generalized quadrangles. Algebr. Groups Geom. 2, 323-354 (1985). Proceedings of the conference on groups and geometry, Part A (Madison, Wis., 1985)
17. Payne, S.E.: An essay on skew translation generalized quadrangles. Geom. Dedic. 32, 93-118 (1989)
18. Payne, S.E.: Finite groups that admit Kantor families. In: Finite Geometries, Groups, and Computation, pp. 191-202. de Gruyter, Berlin (2006)
19. Payne, S.E., Thas, J.A.: Finite Generalized Quadrangles, second edn. EMS Series of Lectures in Mathematics. European Mathematical Society, Zurich (2009)
20. Rostermundt, R.: Elation groups of the Hermitian surface H (3, q2) over a finite field of characteristic
2. Innov. Incid. Geom. 5, 117-128 (2007)
21. Thas, J.A.: Generalized quadrangles and flocks of cones. Eur. J. Comb. 8, 441-452 (1987)
22. Thas, J.A.: Generalized quadrangles of order (s, s2), III. J. Comb. Theory, Ser. A 87, 247-272 (1999)
23. Thas, J.A., Thas, K., Van Maldeghem, H.: Translation Generalized Quadrangles. Series in Pure Mathematics, vol.
26. World Scientific, Hackensack (2006)
24. Thas, K.: Some basic questions and conjectures on elation generalized quadrangles, and their solutions. Bull. Belg. Math. Soc. Simon Stevin 12, 909-918 (2005)
25. Thas, K.: A question of Kantor on elations of dual translation generalized quadrangles. Adv. Geom.
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