Groups whose proper subgroups of infinite rank have a permutability transitive relation

Pub Date : 2024-05-17 DOI:10.1515/jgth-2023-0296

Adolfo Ballester Bolinches, Maria De Falco, Francesco de Giovanni, Carmela Musella

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引用次数: 0

Abstract

Let 𝐺 be a group. A subgroup 𝐻 of 𝐺 is called permutable if H ⁢ X = X ⁢ H

HX=XH

for all subgroups 𝑋 of 𝐺. Permutability is not in general a transitive relation, and 𝐺 is called a PT

\mathrm{PT}

-group if, whenever 𝐾 is a permutable subgroup of 𝐺 and 𝐻 is a permutable subgroup of 𝐾, we always have that 𝐻 is permutable in 𝐺. The property PT

\mathrm{PT}

is not inherited by subgroups, and 𝐺 is called a PT ̄

\overline{\mathrm{PT}}

-group if all its subgroups have the PT

\mathrm{PT}

-property. We prove that if 𝐺 is a soluble group of infinite rank whose proper subgroups of infinite rank have the PT

\mathrm{PT}

-property, then 𝐺 is a PT ̄

\overline{\mathrm{PT}}

-group.

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无穷级的适当子群具有可变传递关系的群

让𝐺是一个群。如果对于𝐺的所有子群𝑋来说，H X = X HX=XH ，那么𝐺的子群𝐻就叫做可周遍群。可周遍性在一般情况下不是一个传递关系，如果，只要𝐺是𝐺的可周遍子群，并且𝐻是𝐺的可周遍子群，我们总是有𝐻在𝐺中是可周遍的，那么𝐺就叫做PT \mathrm{PT} -群。子群不会继承 PT \mathrm{PT} 的属性，如果𝐺 的所有子群都是可变子群，那么𝐺 被称为 PT ̄ \overline\mathrm{PT}} 。 -我们证明，如果𝐺是一个无穷秩的可解群，其无穷秩的适当子群具有 PT \mathrm{PT} -属性，那么𝐺就是一个 PT ̄ \overline\mathrm{PT}} -群。 -群。

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