Finite groups with 𝐻_𝜎-subnormally embedded 𝔑_𝜎-critical subgroups

Abstract 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> \sigma=\{\sigma_{i}\mid i\in 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> \sigma_{i} -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\in 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> \mathfrak{N}_{\sigma} -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> \lvert H\rvert 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> \lvert G:H\rvert 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=H_{0}\leq H_{1}\leq\cdots\leq H_{t}=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> H_{i-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> H_{i} 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> H_{i}/(H_{i-1})_{H_{i}}