A Theorem Concerning Nets Arising from Generalized Quadrangles with a Regular Point |
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Authors: | Koen Thas |
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Institution: | (1) Department of Pure Mathematics and Computer Algebra, Ghent University, Galglaan 2, B-9000 Ghent, Belgium |
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Abstract: | Suppose
is a generalized quadrangle (GQ) of order
, with a regular point. Then there is a net which arises from this regular point. We prove that if such a net has a proper subnet with the same degree as the net, then it must be an affine plane of order t. Also, this affine plane induces a proper subquadrangle of order t containing the regular point, and we necessarily have that
. This result has many applications, of which we give one example. Suppose
is an elation generalized quadrangle (EGQ) of order
, with elation point p. Then
is called a skew translation generalized quadrangle (STGQ) with base-point p if there is a full group of symmetries about p of order t which is contained in the elation group. We show that a GQ
of order s is an STGQ with base-point p if and only if p is an elation point which is regular. |
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Keywords: | generalized quadrangle net subquadrangle symmetries |
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