A character theoretic formula for the base size
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Abstract
Abstract A base for a permutation group G acting on a set $$\Omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Ω</mml:mi> </mml:math> is a sequence $${\mathcal {B}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>B</mml:mi> </mml:math> of points of $$\Omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Ω</mml:mi> </mml:math> such that the pointwise stabiliser $$G_{{\mathcal {B}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>G</mml:mi> <mml:mi>B</mml:mi> </mml:msub> </mml:math> is trivial. The base size of G is the size of a smallest base for G . We derive a character theoretic formula for the base size of a class of groups admitting a certain kind of irreducible character. Moreover, we prove a formula for enumerating the non-equivalent bases for G of size $$l\in {\mathbb {N}}.$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>l</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>N</mml:mi> <mml:mo>.</mml:mo> </mml:mrow> </mml:math> As a consequence of our results, we present a very short, entirely algebraic proof of the formula of Mecenero and Spiga for the base size of the symmetric group $$\textrm{S}_n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mtext>S</mml:mtext> <mml:mi>n</mml:mi> </mml:msub> </mml:math> acting on the k -element subsets of $$\{1,2,3,\ldots ,n\}.$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>{</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mn>3</mml:mn> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo>}</mml:mo> <mml:mo>.</mml:mo> </mml:mrow> </mml:math> Our methods also provide a formula for the base size of many product type permutation groups.
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