On the heritability of the Sylow 
	    
	    
	  -theorem by subgroups

Vdovin, E. P.; Manzaeva, N. Ch.; Ревин, Д. О.

doi:10.1070/sm9185

Cited by 3 publications

(3 citation statements)

References 43 publications

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“…be the set of all overgroups of X-maximal subgroups. The following result, which follows from Theorem 2 and the main result of [16], is not so obvious.…”

Section: Motivation and Historical Remarksmentioning

confidence: 97%

“…In the second case, from Lemma 2 it follows that H ∩ N ∈ Hall X (N ) ⊆ Hall π (N ) and H ∩ N ⩽ K ∩ N ⩽ N . According to Theorem 1.4, [16], K ∩ N ∈ D π . Since any π-Hall subgroup from K ∩ N lies in X, we have K ∩ N ∈ D X .…”

Section: Notation and Preliminary Lemmasmentioning

confidence: 98%

“…The class D X is closed under taking normal subgroups of homomorphic images and extensions, 4 and, in particular, is a Fitting class (see the definition in [15]), and hence in any group there exits the D X -radical. Note that, in general, D X is not a complete class, because it may fail to be closed under taking subgroups (see [16], Theorem 1.7).…”

Section: Motivation and Historical Remarksmentioning

confidence: 99%

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When is the search of relatively maximal subgroups reduced to quotient groups?

Guo

Ревин

2022

Izv. Math.

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Let $\mathfrak{X}$ be a class finite groups closed under taking subgroups, homomorphic images, and extensions, and let $\mathrm{k}_{\mathfrak{X}}(G)$ be the number of conjugacy classes $\mathfrak{X}$-maximal subgroups of a finite group $G$. The natural problem calling for a description, up to conjugacy, of the $\mathfrak{X}$-maximal subgroups of a given finite group is not inductive. In particular, generally speaking, the image of an $\mathfrak{X}$-maximal subgroup is not $\mathfrak{X}$-maximal in the image of a homomorphism. Nevertheless, there exist group homomorphisms that preserve the number of conjugacy classes of maximal $\mathfrak{X}$-subgroups (for example, the homomorphisms whose kernels are $\mathfrak{X}$-groups). Under such homomorphisms, the image of an $\mathfrak{X}$-maximal subgroup is always $\mathfrak{X}$-maximal, and, moreover, there is a natural bijection between the conjugacy classes of $\mathfrak{X}$-maximal subgroups of the image and preimage. In the present paper, all such homomorphisms are completely described. More precisely, it is shown that, for a homomorphism $\phi$ from a group $G$, the equality $\mathrm{k}_{\mathfrak{X}}(G)=\mathrm{k}_{\mathfrak{X}}(\operatorname{im} \phi)$ holds if and only if $\mathrm{k}_{\mathfrak{X}}(\ker \phi)=1$, which in turn is equivalent to the fact that the composition factors of the kernel of $\phi$ lie in an explicitly given list.

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“…be the set of all overgroups of X-maximal subgroups. The following result, which follows from Theorem 2 and the main result of [16], is not so obvious.…”

Section: Motivation and Historical Remarksmentioning

confidence: 97%

Section: Notation and Preliminary Lemmasmentioning

confidence: 98%

Section: Motivation and Historical Remarksmentioning

confidence: 99%

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When is the search of relatively maximal subgroups reduced to quotient groups?

Guo

Ревин

2022

Izv. Math.

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The reduction theorem for relatively maximal subgroups

2021

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Let [Formula: see text] be a class of finite groups closed under taking subgroups, homomorphic images and extensions. It is known that if [Formula: see text] is a normal subgroup of a finite group [Formula: see text] then the image of an [Formula: see text]-maximal subgroup [Formula: see text] of [Formula: see text] in [Formula: see text] is not, in general, [Formula: see text]-maximal in [Formula: see text]. We say that the reduction [Formula: see text]-theorem holds for a finite group [Formula: see text] if, for every finite group [Formula: see text] that is an extension of [Formula: see text] (i.e. contains [Formula: see text] as a normal subgroup), the number of conjugacy classes of [Formula: see text]-maximal subgroups in [Formula: see text] and [Formula: see text] is the same. The reduction [Formula: see text]-theorem for [Formula: see text] implies that [Formula: see text] is [Formula: see text]-maximal in [Formula: see text] for every extension [Formula: see text] of [Formula: see text] and every [Formula: see text]-maximal subgroup [Formula: see text] of [Formula: see text]. In this paper, we prove that the reduction [Formula: see text]-theorem holds for [Formula: see text] if and only if all [Formula: see text]-maximal subgroups of [Formula: see text] are conjugate in [Formula: see text] and classify the finite groups with this property in terms of composition factors.

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When is the search of relatively maximal subgroups reduced to quotient groups?

Guo

Ревин³

2022

Известия Российской академии наук. Серия математическая

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Пусть $\mathfrak{X}$ - класс конечных групп, замкнутый относительно подгрупп, гомоморфных образов и расширений, и $\mathrm{k}_{\mathfrak{X}}(G)$ - число классов сопряженности $\mathfrak{X}$-максимальных подгрупп конечной группы $G$. Естественная задача - описать с точностью до сопряженности $\mathfrak{X}$-максимальные подгруппы данной конечной группы - не индуктивна. В частности, в образе гомоморфизма образ $\mathfrak{X}$-максимальной подгруппы, вообще говоря, не $\mathfrak{X}$-максимален. Существуют гомоморфизмы, сохраняющие число классов сопряженности максимальных $\mathfrak{X}$-подгрупп (например, гомоморфизмы, ядра которых - $\mathfrak{X}$-группы). Относительно таких гомоморфизмов образ $\mathfrak{X}$-максимальной подгруппы всегда $\mathfrak{X}$-максимален и существует естественная биекция между классами сопряженности $\mathfrak{X}$-максимальных подгрупп образа и прообраза. В работе такие гомоморфизмы полностью описаны. Доказано, что для гомоморфизма $\phi$ из группы $G$ равенство $\mathrm{k}_{\mathfrak{X}}(G)=\mathrm{k}_{\mathfrak{X}}(\operatorname{im} \phi)$ выполнено, если и только если $\mathrm{k}_{\mathfrak{X}}(\ker \phi)=1$, а это, в свою очередь, равносильно тому, что композиционные факторы ядра $\phi$ принадлежат известному списку. Библиография: 25 наименований.

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On the heritability of the Sylow -theorem by subgroups

Abstract: Let be a set of primes. We say that the Sylow -theorem holds for a finite group , or is a -group, if the maximal -subgroups of … Show more

Cited by 3 publications

References 43 publications

When is the search of relatively maximal subgroups reduced to quotient groups?

When is the search of relatively maximal subgroups reduced to quotient groups?

The reduction theorem for relatively maximal subgroups

When is the search of relatively maximal subgroups reduced to quotient groups?

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