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Automorphisms of a Generalized Quadrangle of Order 6
Pesak, Ryan
Pesak, Ryan
Abstract
In this thesis, we study the symmetries of the putative generalized quadrangle of order 6. Although it is unknown whether such a quadrangle Q can exist, we show that if it does, that Q cannot be transitive on either points or lines. We first cover the background necessary for studying this problem. Namely, the theory of groups and group actions, the theory of generalized quadrangles, and automorphisms of GQs. We then prove that a generalized quadrangle Q of order 6 cannot have a point- or line-transitive automorphism group, and we also prove that if a group G acts faithfully on Q such that 259 | |G|, then G is not solvable. Along the way, we develop techniques for studying composite order automorphisms of a generalized quadrangle. Specifically, we deal with automorphisms of order pk and pq, where p and q are prime.
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Date
2023-05-01
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Swartz, Eric
Clare, Pierre
Gert, Joshua
Clare, Pierre
Gert, Joshua
Department
Mathematics