<p>In this paper, we establish some results that confirm that <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>σ</mi> </math></EquationSource> </InlineEquation>-normal and <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>σ</mi> </math></EquationSource> </InlineEquation>-permutable subgroups play an important role in the structural analysis of finite groups associated with <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>σ</mi> </math></EquationSource> </InlineEquation>-partitions of the set of all primes. We show that the difference between these two subgroup embedding properties and their corresponding associated classes in the <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>σ</mi> </math></EquationSource> </InlineEquation>-soluble universe is just the Hall <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>σ</mi> </math></EquationSource> </InlineEquation>-structure. Furthermore, we prove that the projectors of every <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>σ</mi> </math></EquationSource> </InlineEquation>-soluble group associated with the saturated formation of all <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$\sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>σ</mi> </math></EquationSource> </InlineEquation>-nilpotent groups are conjugate and coincide with the covering subgroups. Some relations between these projectors and the <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$\sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>σ</mi> </math></EquationSource> </InlineEquation>-nilpotent and <InlineEquation ID="IEq15"> <EquationSource Format="TEX">$\sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>σ</mi> </math></EquationSource> </InlineEquation>-self-normalising subgroups of a <InlineEquation ID="IEq16"> <EquationSource Format="TEX">$\sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>σ</mi> </math></EquationSource> </InlineEquation>-soluble group are also exhibited.</p>

On $\sigma $-Normal Subgroups of Finite $\sigma $-Soluble Groups

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2026年7月3日

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A. A. Heliel,
M. M. Al-Shomrani,
A. Ballester-Bolinches

摘要

In this paper, we establish some results that confirm that $\sigma $ $σ$ -normal and $\sigma $ $σ$ -permutable subgroups play an important role in the structural analysis of finite groups associated with $\sigma $ $σ$ -partitions of the set of all primes. We show that the difference between these two subgroup embedding properties and their corresponding associated classes in the $\sigma $ $σ$ -soluble universe is just the Hall $\sigma $ $σ$ -structure. Furthermore, we prove that the projectors of every $\sigma $ $σ$ -soluble group associated with the saturated formation of all $\sigma $ $σ$ -nilpotent groups are conjugate and coincide with the covering subgroups. Some relations between these projectors and the $\sigma $ $σ$ -nilpotent and $\sigma $ $σ$ -self-normalising subgroups of a $\sigma $ $σ$ -soluble group are also exhibited.

On \(\sigma \)-Normal Subgroups of Finite \(\sigma \)-Soluble Groups

摘要