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A character theoretic formula for the base size.

Archiv der Mathematik. Archives of mathematics. Archives mathematiques·2025
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Another character theoretic formula for base size.

Coen Del Valle1

  • 1School of Mathematics and Statistics, The Open University, Milton Keynes, UK.

Archiv Der Mathematik. Archives of Mathematics. Archives Mathematiques
|October 13, 2025
PubMed
Summary
This summary is machine-generated.

This study proves a conjecture on permutation group bases and derives new formulas for base size. These findings advance the understanding of permutation group properties and their applications.

Keywords:
Base sizeIrreducible characterKülshammer graph

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Area of Science:

  • Group Theory
  • Abstract Algebra
  • Combinatorics

Background:

  • Permutation groups acting on sets are fundamental in abstract algebra.
  • The base size of a permutation group is a key invariant, with applications in computational group theory.
  • A 2013 conjecture by Fritzsche, Külshammer, and Reiche focused on determining this base size.

Purpose of the Study:

  • To prove the 2013 conjecture regarding the base size of permutation groups.
  • To generalize the conjecture and derive new formulas for base size.
  • To obtain an alternative character theoretic formula for the base size of specific permutation groups.

Main Methods:

  • Extending previous results on permutation group bases.
  • Utilizing character theory to derive new formulas.
  • Analyzing the base size of the symmetric group acting on subsets.

Main Results:

  • The 2013 conjecture by Fritzsche, Külshammer, and Reiche has been proven.
  • A generalized conjecture and new formulas for base size have been derived.
  • An alternative character theoretic formula for the base size of certain permutation groups is established.

Conclusions:

  • The study successfully resolves a significant conjecture in permutation group theory.
  • New theoretical tools and formulas are provided for calculating base sizes.
  • The findings offer a deeper understanding of permutation group structure and properties.