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Isomorphisms of groups related to flocks

Koen Thas (UGent)
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Abstract
A truly fruitful way to construct finite generalized quadrangles is through the detection of Kantor families in the general 5-dimensional Heisenberg group over some finite field . 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 . In this paper, we resolve the question of Payne for the complete class . In fact we do more-we show that flock quadrangles are characterized by their groups. Several (sometimes surprising) by-products are described in both odd and even characteristic.
Keywords
Flock quadrangle, Elation quadrangle, Automorphism group, Heisenberg group, Characterization, ELATION GENERALIZED QUADRANGLES, QUADRICS, OVOIDS, SETS, PRIME

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MLA
Thas, Koen. “Isomorphisms of Groups Related to Flocks.” JOURNAL OF ALGEBRAIC COMBINATORICS 36.1 (2012): 111–121. Print.
APA
Thas, K. (2012). Isomorphisms of groups related to flocks. JOURNAL OF ALGEBRAIC COMBINATORICS, 36(1), 111–121.
Chicago author-date
Thas, Koen. 2012. “Isomorphisms of Groups Related to Flocks.” Journal of Algebraic Combinatorics 36 (1): 111–121.
Chicago author-date (all authors)
Thas, Koen. 2012. “Isomorphisms of Groups Related to Flocks.” Journal of Algebraic Combinatorics 36 (1): 111–121.
Vancouver
1.
Thas K. Isomorphisms of groups related to flocks. JOURNAL OF ALGEBRAIC COMBINATORICS. 2012;36(1):111–21.
IEEE
[1]
K. Thas, “Isomorphisms of groups related to flocks,” JOURNAL OF ALGEBRAIC COMBINATORICS, vol. 36, no. 1, pp. 111–121, 2012.
@article{7183352,
  abstract     = {A truly fruitful way to construct finite generalized quadrangles is through the detection of Kantor families in the general 5-dimensional Heisenberg group over some finite field . 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 . 
In this paper, we resolve the question of Payne for the complete class . In fact we do more-we show that flock quadrangles are characterized by their groups. 
Several (sometimes surprising) by-products are described in both odd and even characteristic.},
  author       = {Thas, Koen},
  issn         = {0925-9899},
  journal      = {JOURNAL OF ALGEBRAIC COMBINATORICS},
  keywords     = {Flock quadrangle,Elation quadrangle,Automorphism group,Heisenberg group,Characterization,ELATION GENERALIZED QUADRANGLES,QUADRICS,OVOIDS,SETS,PRIME},
  language     = {eng},
  number       = {1},
  pages        = {111--121},
  title        = {Isomorphisms of groups related to flocks},
  url          = {http://dx.doi.org/10.1007/s10801-011-0326-0},
  volume       = {36},
  year         = {2012},
}

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