Finite groups with 𝐻𝜎-subnormally embedded 𝔑𝜎-critical subgroups

Let 𝐺 be a finite group and <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\} a partition of the set of all primes. The group 𝐺 is said to be 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 <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 , 𝜎-nilpotent if 𝐺 is a direct product of 𝜎-primary groups, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi mathvariant="fraktur">N</m:mi> <m:mi>σ</m:mi> </m:msub> </m:math> Nσ -critical if 𝐺 is not 𝜎-nilpotent but all proper subgroups of 𝐺 are 𝜎-nilpotent. A subgroup 𝐻 of 𝐺 is called a 𝜎-Hall subgroup of 𝐺 provided <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">|</m:mo> <m:mi>H</m:mi> <m:mo stretchy="false">|</m:mo> </m:mrow> </m:math> H and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">|</m:mo> <m:mrow> <m:mi>G</m:mi> <m:mo lspace="0.278em" rspace="0.278em">:</m:mo> <m:mi>H</m:mi> </m:mrow> <m:mo stretchy="false">|</m:mo> </m:mrow> </m:math> G:H are 𝜎-coprime and 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>H</m:mi> <m:mo>=</m:mo> <m:msub> <m:mi>H</m:mi> <m:mn>0</m:mn> </m:msub> <m:mo>≤</m:mo> <m:msub> <m:mi>H</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>H</m:mi> <m:mi>t</m:mi> </m:msub> <m:mo>=</m:mo> <m:mi>G</m:mi> </m:mrow> </m:math> H=H₀≤ H₁≤⋯≤ Hₜ=G such that either <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>H</m:mi> <m:mrow> <m:mi>i</m:mi> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:math> Hᵢ₋₁ is normal in <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 or <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>H</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>H</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>H</m:mi> <m:mi>i</m:mi> </m:msub> </m:msub> </m:mrow> </m:math> Hᵢ/(Hᵢ₋₁)_{Hᵢ} 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>t</m:mi> </m:mrow> </m:mrow> </m:math> i=1,…,t . A subgroup 𝐻 of 𝐺 is called <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>H</m:mi> <m:mi>σ</m:mi> </m:msub> </m:math> Hσ -subnormally embedded in 𝐺 if 𝐻 is a 𝜎-Hall subgroup of some 𝜎-subnormal subgroup of 𝐺. In this paper, we prove that the second derived subgroup of a finite group 𝐺 is 𝜎-nilpotent if every <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi mathvariant="fraktur">N</m:mi> <m:mi>σ</m:mi> </m:msub> </m:math> Nσ -critical subgroup of 𝐺 is <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>H</m:mi> <m:mi>σ</m:mi> </m:msub> </m:math> Hσ -subnormally embedded in 𝐺.