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February 12, 2026Bulletin of the London Mathematical Society0 citationsOpen Access

Sets preserved by a large subgroup of the special linear group

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TPThang PhamLQLe Quang‐HungKSKaloyan Slavov

Key Points

  • The aim is to establish a bound on the size of subgroups of the special linear group that preserve a given set in the affine plane over a finite field.
  • Utilized combinatorial arguments
  • Applied point-line incidence bounds from prior work by Mockenhaupt and Tao (2004)
  • Analyzed subsets of the affine plane
  • If a set is preserved by more than a certain number of subgroup elements, it must lie on a line through the origin.
  • The result is sharp, confirming exact bounds in general cases.

Abstract

Abstract Let be a subset of the affine plane over a finite field . We bound the size of the subgroup of that preserves . Specifically, there exists a constant such that for any , if is preserved by more than elements of , then is contained in a line through the origin. This result is sharp in general, and will be proved by using combinatorial arguments and applying a point‐line incidence bound in due to Mockenhaupt and Tao (2004).

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Cite This Study

Pham et al. (2026) studied this question.

synapsesocial.com/papers/698d6f0d5be6419ac0d5519fhttps://doi.org/10.1112/blms.70298
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