### Abstract

A generalized quadrangle is a point-line incidence geometry Q such that: (i) any two points lie on at most one line, and (ii) given a line ℓ and a point P not incident with ℓ, there is a unique point of ℓ collinear with P. The finite Moufang generalized quadrangles were classified by Fong and Seitz [Invent. Math. 21 (1973), 1-57; Invent. Math. 24 (1974), 191-239], and we study a larger class of generalized quadrangles: the antiflag-transitive quadrangles. An antiflag of a generalized quadrangle is a nonincident point-line pair (P, ℓ), and we say that the generalized quadrangle Q is antiflag-transitive if the group of collineations is transitive on the set of all antiflags. We prove that if a finite thick generalized quadrangle Q is antiflag-transitive, then Q is either a classical generalized quadrangle or is the unique generalized quadrangle of order (3, 5) or its dual. Our approach uses the theory of locally s-arc-transitive graphs developed by Giudici, Li, and Praeger [Trans. Amer. Math. Soc. 356 (2004), 291-317] to characterize antiflag-transitive generalized quadrangles and then the work of Alavi and Burness [J. Algebra 421 (2015), 187-233] on “large” subgroups of simple groups of Lie type to fully classify them.

Original language | English |
---|---|

Pages (from-to) | 1551-1601 |

Number of pages | 51 |

Journal | Transactions of the American Mathematical Society |

Volume | 370 |

Issue number | 3 |

DOIs | |

Publication status | Published - 1 Mar 2018 |

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*Transactions of the American Mathematical Society*,

*370*(3), 1551-1601. https://doi.org/10.1090/tran/6984

}

*Transactions of the American Mathematical Society*, vol. 370, no. 3, pp. 1551-1601. https://doi.org/10.1090/tran/6984

**A classification of finite antiflag-transitive generalized quadrangles.** / Bamberg, John; Li, Cai Heng; Swartz, Eric.

Research output: Contribution to journal › Article

TY - JOUR

T1 - A classification of finite antiflag-transitive generalized quadrangles

AU - Bamberg, John

AU - Li, Cai Heng

AU - Swartz, Eric

PY - 2018/3/1

Y1 - 2018/3/1

N2 - A generalized quadrangle is a point-line incidence geometry Q such that: (i) any two points lie on at most one line, and (ii) given a line ℓ and a point P not incident with ℓ, there is a unique point of ℓ collinear with P. The finite Moufang generalized quadrangles were classified by Fong and Seitz [Invent. Math. 21 (1973), 1-57; Invent. Math. 24 (1974), 191-239], and we study a larger class of generalized quadrangles: the antiflag-transitive quadrangles. An antiflag of a generalized quadrangle is a nonincident point-line pair (P, ℓ), and we say that the generalized quadrangle Q is antiflag-transitive if the group of collineations is transitive on the set of all antiflags. We prove that if a finite thick generalized quadrangle Q is antiflag-transitive, then Q is either a classical generalized quadrangle or is the unique generalized quadrangle of order (3, 5) or its dual. Our approach uses the theory of locally s-arc-transitive graphs developed by Giudici, Li, and Praeger [Trans. Amer. Math. Soc. 356 (2004), 291-317] to characterize antiflag-transitive generalized quadrangles and then the work of Alavi and Burness [J. Algebra 421 (2015), 187-233] on “large” subgroups of simple groups of Lie type to fully classify them.

AB - A generalized quadrangle is a point-line incidence geometry Q such that: (i) any two points lie on at most one line, and (ii) given a line ℓ and a point P not incident with ℓ, there is a unique point of ℓ collinear with P. The finite Moufang generalized quadrangles were classified by Fong and Seitz [Invent. Math. 21 (1973), 1-57; Invent. Math. 24 (1974), 191-239], and we study a larger class of generalized quadrangles: the antiflag-transitive quadrangles. An antiflag of a generalized quadrangle is a nonincident point-line pair (P, ℓ), and we say that the generalized quadrangle Q is antiflag-transitive if the group of collineations is transitive on the set of all antiflags. We prove that if a finite thick generalized quadrangle Q is antiflag-transitive, then Q is either a classical generalized quadrangle or is the unique generalized quadrangle of order (3, 5) or its dual. Our approach uses the theory of locally s-arc-transitive graphs developed by Giudici, Li, and Praeger [Trans. Amer. Math. Soc. 356 (2004), 291-317] to characterize antiflag-transitive generalized quadrangles and then the work of Alavi and Burness [J. Algebra 421 (2015), 187-233] on “large” subgroups of simple groups of Lie type to fully classify them.

UR - http://www.scopus.com/inward/record.url?scp=85039798163&partnerID=8YFLogxK

U2 - 10.1090/tran/6984

DO - 10.1090/tran/6984

M3 - Article

VL - 370

SP - 1551

EP - 1601

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 3

ER -