On 𝜎-permutable subgroups of 𝜎-soluble finite groups

Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>σ</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:msub> <m:mi>σ</m:mi> <m:mi>i</m:mi> </m:msub> <m:mo fence="true" lspace="0em" rspace="0em">∣</m:mo> <m:mrow> <m:mi>i</m:mi> <m:mo>∈</m:mo> <m:mi>I</m:mi> </m:mrow> <m:mo stretchy="false">}</m:mo> </m:mrow> </m:mrow> </m:math> σ=\{σᵢ i∈ I\} be some partition of the set of all primes and 𝐺 a finite group. Then 𝐺 is said to be 𝜎-full if 𝐺 has a Hall <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>σ</m:mi> <m:mi>i</m:mi> </m:msub> </m:math> σᵢ -subgroup for all <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>i</m:mi> <m:mo>∈</m:mo> <m:mi>I</m:mi> </m:mrow> </m:math> i∈ I and 𝜎-primary if 𝐺 is a <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>σ</m:mi> <m:mi>i</m:mi> </m:msub> </m:math> σᵢ -group for some 𝑖. In addition, 𝐺 is 𝜎-soluble if every chief factor of 𝐺 is 𝜎-primary and 𝜎-nilpotent if 𝐺 is a direct product of 𝜎-primary groups. We write <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>G</m:mi> <m:msub> <m:mi mathvariant="fraktur">N</m:mi> <m:mi>σ</m:mi> </m:msub> </m:msup> </m:math> G^{Nσ} for the 𝜎-nilpotent residual of 𝐺, which is the intersection of all normal subgroups 𝑁 of 𝐺 with 𝜎-nilpotent <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>G</m:mi> <m:mo>/</m:mo> <m:mi>N</m:mi> </m:mrow> </m:math> G/N . A subgroup 𝐴 of 𝐺 is said to be 𝜎-permutable in 𝐺 provided 𝐺 is 𝜎-full and 𝐴 permutes with all Hall <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>σ</m:mi> <m:mi>i</m:mi> </m:msub> </m:math> σᵢ -subgroups 𝐻 of 𝐺 (that is, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>A</m:mi> <m:mo>⁢</m:mo> <m:mi>H</m:mi> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:mi>H</m:mi> <m:mo>⁢</m:mo> <m:mi>A</m:mi> </m:mrow> </m:mrow> </m:math> AH=HA ) for all 𝑖. And 𝐴 is 𝜎-subnormal in 𝐺 if there is a subgroup chain <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>A</m:mi> <m:mo>=</m:mo> <m:msub> <m:mi>A</m:mi> <m:mn>0</m:mn> </m:msub> <m:mo>≤</m:mo> <m:msub> <m:mi>A</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo>≤</m:mo> <m:mi mathvariant="normal">⋯</m:mi> <m:mo>≤</m:mo> <m:msub> <m:mi>A</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo>=</m:mo> <m:mi>G</m:mi> </m:mrow> </m:math> A=A₀≤ A₁≤⋯≤ Aₙ=G such that either <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>A</m:mi> <m:mrow> <m:mi>i</m:mi> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo>⁢</m:mo> <m:mi mathvariant="normal">⊴</m:mi> <m:mo>⁢</m:mo> <m:msub> <m:mi>A</m:mi> <m:mi>i</m:mi> </m:msub> </m:mrow> </m:math> Aᵢ₋₁ Aᵢ or <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>A</m:mi> <m:mi>i</m:mi> </m:msub> <m:mo>/</m:mo> <m:msub> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>A</m:mi> <m:mrow> <m:mi>i</m:mi> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:msub> <m:mi>A</m:mi> <m:mi>i</m:mi> </m:msub> </m:msub> </m:mrow> </m:math> Aᵢ/(Aᵢ₋₁)_{Aᵢ} is 𝜎-primary for all <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>i</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mi mathvariant="normal">…</m:mi> <m:mo>,</m:mo> <m:mi>n</m:mi> </m:mrow> </m:mrow> </m:math> i=1,…,n . We prove that if 𝐺 is a 𝜎-soluble group, then 𝜎-permutability is a transitive relation in 𝐺 if and only if <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:msup> <m:mi>G</m:mi> <m:msub> <m:mi mathvariant="fraktur">N</m:mi> <m:mi>σ</m:mi> </m:msub> </m:msup> <m:mo>∩</m:mo> <m:msup> <m:mi>A</m:mi> <m:mi>G</m:mi> </m:msup> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:msup> <m:mi>G</m:mi> <m:msub> <m:mi mathvariant="fraktur">N</m:mi> <m:mi>σ</m:mi> </m:msub> </m:msup> <m:mo>∩</m:mo> <m:msub> <m:mi>A</m:mi> <m:mi>G</m:mi> </m:msub> </m:mrow> </m:mrow> </m:math> G^{Nσ}∩ AG=G^{Nσ}∩ AG for every 𝜎-subnormal subgroup 𝐴 of 𝐺.