𝑝-parts of stabilizers in primitive permutation groups

Abstract Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo>,</m:mo> <m:mi mathvariant="normal">Ω</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> (G,\Omega) be a finite primitive permutation group. Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>p</m:mi> <m:mn>2</m:mn> </m:msup> </m:math> p^{2} divide <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">|</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">|</m:mo> </m:mrow> </m:math> \lvert G\rvert , for a prime 𝑝. We show that, when 𝐺 is solvable, there exists a subset of Ω whose stabilizer 𝑆 has the property that <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mn>1</m:mn> <m:mo>&lt;</m:mo> <m:msub> <m:mrow> <m:mo stretchy="false">|</m:mo> <m:mi>S</m:mi> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mi>p</m:mi> </m:msub> <m:mo>&lt;</m:mo> <m:msub> <m:mrow> <m:mo stretchy="false">|</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mi>p</m:mi> </m:msub> </m:mrow> </m:math> 1&lt;\lvert S\rvert_{p}&lt;\lvert G\rvert_{p} . We offer a counting argument which should be helpful when 𝐺 is not solvable.