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• W4289333405 abstract "Abstract In this paper, 𝐺 is a finite group and 𝜎 a partition of the set of all primes ℙ, that is, <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>∣</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 iin I} , where <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi mathvariant=double-struck>P</m:mi> <m:mo>=</m:mo> <m:mrow> <m:msub> <m:mo largeop=true symmetric=true>⋃</m:mo> <m:mrow> <m:mi>i</m:mi> <m:mo>∈</m:mo> <m:mi>I</m:mi> </m:mrow> </m:msub> <m:msub> <m:mi>σ</m:mi> <m:mi>i</m:mi> </m:msub> </m:mrow> </m:mrow> </m:math> mathbb{P}=bigcup_{iin I}sigma_{i} and <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mrow> <m:msub> <m:mi>σ</m:mi> <m:mi>i</m:mi> </m:msub> <m:mo>∩</m:mo> <m:msub> <m:mi>σ</m:mi> <m:mi>j</m:mi> </m:msub> </m:mrow> <m:mo>=</m:mo> <m:mi mathvariant=normal>∅</m:mi> </m:mrow> </m:math> sigma_{i}capsigma_{j}=emptyset 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>j</m:mi> </m:mrow> </m:math> ineq j . If 𝑛 is an integer, we write <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mrow> <m:mi>σ</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi>n</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> <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>∣</m:mo> <m:mrow> <m:mrow> <m:msub> <m:mi>σ</m:mi> <m:mi>i</m:mi> </m:msub> <m:mo>∩</m:mo> <m:mrow> <m:mi>π</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi>n</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> </m:mrow> <m:mo>≠</m:mo> <m:mi mathvariant=normal>∅</m:mi> </m:mrow> <m:mo stretchy=false>}</m:mo> </m:mrow> </m:mrow> </m:math> sigma(n)={sigma_{i}midsigma_{i}cappi(n)neqemptyset} and <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mrow> <m:mi>σ</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:mrow> <m:mi>σ</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mrow> <m:mo fence=true stretchy=false>|</m:mo> <m:mi>G</m:mi> <m:mo fence=true stretchy=false>|</m:mo> </m:mrow> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> sigma(G)=sigma(lvert Grvert) . A 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:mrow> <m:mi>i</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:mrow> </m:math> i=i(G) and 𝜎-soluble if every chief factor of 𝐺 is 𝜎-primary. We say that 𝐺 is a 𝜎-tower group if either <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi>G</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> G=1 or 𝐺 has a normal series <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mn>1</m:mn> <m:mo>=</m:mo> <m:msub> <m:mi>G</m:mi> <m:mn>0</m:mn> </m:msub> <m:mo><</m:mo> <m:msub> <m:mi>G</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>G</m:mi> <m:mrow> <m:mi>t</m:mi> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo><</m:mo> <m:msub> <m:mi>G</m:mi> <m:mi>t</m:mi> </m:msub> <m:mo>=</m:mo> <m:mi>G</m:mi> </m:mrow> </m:math> 1=G_{0}<G_{1}<cdots<G_{t-1}<G_{t}=G such that <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:msub> <m:mi>G</m:mi> <m:mi>i</m:mi> </m:msub> <m:mo>/</m:mo> <m:msub> <m:mi>G</m:mi> <m:mrow> <m:mi>i</m:mi> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:mrow> </m:math> G_{i}/G_{i-1} 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, <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:msub> <m:mi>σ</m:mi> <m:mi>i</m:mi> </m:msub> <m:mo>∈</m:mo> <m:mrow> <m:mi>σ</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:mrow> </m:math> sigma_{i}insigma(G) , and <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi>G</m:mi> <m:mo>/</m:mo> <m:msub> <m:mi>G</m:mi> <m:mi>i</m:mi> </m:msub> </m:mrow> </m:math> G/G_{i} and <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:msub> <m:mi>G</m:mi> <m:mrow> <m:mi>i</m:mi> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:math> G_{i-1} are <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:msubsup> <m:mi>σ</m:mi> <m:mi>i</m:mi> <m:mo>′</m:mo> </m:msubsup> </m:math> sigma_{i}^{prime} -groups 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,ldots,t . A subgroup 𝐴 of 𝐺 is said to be 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>t</m:mi> </m:msub> <m:mo>=</m:mo> <m:mi>G</m:mi> </m:mrow> </m:math> A=A_{0}leq A_{1}leqcdotsleq A_{t}=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> A_{i-1}trianglelefteq A_{i} 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> A_{i}/(A_{i-1})_{A_{i}} 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,ldots,t . In this paper, answering to Question 4.8 in [A. N. Skiba, On 𝜎-subnormal and 𝜎-permutable subgroups of finite groups, J. Algebra 436 (2015), 1–16], we prove that a 𝜎-soluble group <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi>G</m:mi> <m:mo>≠</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> Gneq 1 with <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mrow> <m:mo fence=true stretchy=false>|</m:mo> <m:mrow> <m:mi>σ</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 fence=true stretchy=false>|</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mi>n</m:mi> </m:mrow> </m:math> lvertsigma(G)rvert=n is a 𝜎-tower group if each of its <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mrow> <m:mi>n</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:math> (n+1) -maximal subgroups is 𝜎-subnormal in 𝐺." @default.
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• W4289333405 date "2022-08-02" @default.
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• W4289333405 title "On finite 𝜎-tower groups" @default.
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