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Another character theoretic formula for base size.

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

Coen Del Valle1

  • 1School of Mathematics and Statistics, University of St Andrews, St Andrews, UK.

Archiv Der Mathematik. Archives of Mathematics. Archives Mathematiques
|April 14, 2025
PubMed
Summary
This summary is machine-generated.

This study introduces a character-theoretic formula for calculating the base size of permutation groups. It also provides a new algebraic proof for the symmetric group and formulas for product type groups.

Keywords:
Base sizeIrreducible characterLarge base

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Area of Science:

  • Group Theory
  • Computational Algebra
  • Combinatorics

Background:

  • Permutation groups are fundamental in abstract algebra.
  • The base size of a permutation group is a key invariant.
  • Understanding base sizes is crucial for group classification and computational applications.

Purpose of the Study:

  • To derive a character-theoretic formula for the base size of specific permutation groups.
  • To develop a method for enumerating non-equivalent bases.
  • To provide an algebraic proof for the base size of the symmetric group and generalize to product type groups.

Main Methods:

  • Character theory applied to permutation group actions.
  • Derivation of novel formulas for base size calculation.
  • Algebraic manipulation and group-theoretic proofs.

Main Results:

  • A character-theoretic formula for the base size of a class of groups.
  • A formula for enumerating non-equivalent bases of a given size.
  • A concise algebraic proof of Mecenero and Spiga's formula for the symmetric group.

Conclusions:

  • The derived formulas offer new insights into the structure of permutation groups.
  • The methods provide efficient tools for computing base sizes.
  • The results generalize to broader classes of permutation groups, including product types.