On Gluck’s conjecture for wreath product type groups

A well-known conjecture of Gluck claims that <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mo stretchy="false">|</m:mo> <m:mi>G</m:mi> <m:mo lspace="0.278em" rspace="0.278em">:</m:mo> <m:mi mathvariant="bold">F</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mo>≤</m:mo> <m:mi>b</m:mi> <m:msup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mn>2</m:mn> </m:msup> </m:mrow> </m:math> G:F(G)≤ b(G)² for all finite solvable groups 𝐺, where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="bold">F</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> F(G) is the Fitting subgroup and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>b</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> b(G) is the largest degree of a complex irreducible character of 𝐺. In this paper, we prove that Gluck’s conjecture holds for all wreath product type groups of the form <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>G</m:mi> <m:mo lspace="0.222em" rspace="0.222em">≀</m:mo> <m:msub> <m:mi>H</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo lspace="0.222em" rspace="0.222em">≀</m:mo> <m:msub> <m:mi>H</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo lspace="0.222em" rspace="0.222em">≀</m:mo> <m:mi mathvariant="normal">⋯</m:mi> <m:mo lspace="0.222em" rspace="0.222em">≀</m:mo> <m:msub> <m:mi>H</m:mi> <m:mi>r</m:mi> </m:msub> </m:mrow> </m:math> G H₁ H₂⋯ Hᵣ , where 𝐺 is a finite solvable group acting primitively on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mrow> <m:mi mathvariant="bold">F</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>/</m:mo> <m:mi mathvariant="normal">Φ</m:mi> </m:mrow> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> F(G)/Φ(G) , and each <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>H</m:mi> <m:mi>i</m:mi> </m:msub> </m:math> Hᵢ is a solvable primitive permutation group of finite degree.