Hall subgroups of finite groups

Ревин, Д. О.; Vdovin, Evgenii Petrovitch

doi:10.1090/conm/402/07585

Cited by 36 publications

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“…In [6,Theorem 7.7] we proved the following theorem. By using this theorem we obtain that a finite group G satisfies D p if and only if every composition factor satisfies D p .…”

Section: Introductionmentioning

confidence: 95%

“…A finite group is said to be a K-group, if all of its composition factors are known simple groups, i.e., are either cyclic of prime order, or alternating, or of Lie type, or sporadic. If we replace the term 'finite group' by the term 'finite K-group', then all results remain valid without the use of the classification of finite simple groups.In [6, Theorem 7.7] we proved the following theorem. …”

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An existence criterion for Hall subgroups of finite groups

Ревин¹,

Vdovin²

2011

Journal of Group Theory

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Abstract. We obtain an existence criterion for Hall subgroups of finite groups in terms of a composition series. As a corollary we provide a solution to Problem 5.65 from the Kourovka notebook. IntroductionLet p be a set of primes. We denote by p 0 the set of all primes not in p, by pðnÞ the set of prime divisors of a positive integer n, for a finite group G we denote pðjGjÞ by pðGÞ. A positive integer n with pðnÞ J p is called a p-number, a group G with pðGÞ J p is called a p-group. A subgroup H of G is called a p-Hall subgroup, if pðHÞ J p and pðjG : HjÞ J p 0 . According to [3] we say that G satisfies E p (or briefly G A E p ), if G contains a p-Hall subgroup. If G A E p and every two p-Hall subgroups are conjugate, then we say that G satisfies C p ðG A C p Þ. If G A C p and each p-subgroup of G is contained in a p-Hall subgroup of G, then we say that G satisfies All results of the paper depend on the classification of finite simple groups. We can avoid its use with the following definition. A finite group is said to be a K-group, if all of its composition factors are known simple groups, i.e., are either cyclic of prime order, or alternating, or of Lie type, or sporadic. If we replace the term 'finite group' by the term 'finite K-group', then all results remain valid without the use of the classification of finite simple groups.In [6, Theorem 7.7] we proved the following theorem.

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“…In [6,Theorem 7.7] we proved the following theorem. By using this theorem we obtain that a finite group G satisfies D p if and only if every composition factor satisfies D p .…”

Section: Introductionmentioning

confidence: 95%

mentioning

confidence: 94%

An existence criterion for Hall subgroups of finite groups

Ревин¹,

Vdovin²

2011

Journal of Group Theory

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“…Классификация π-холловых подгрупп в группах лиева типа в случае, когда 2, p / ∈ π, получена в [42]- [44], [74]. Случай, когда 2 ∈ π, а 3, p / ∈ π, разобран в [45] для линейных и симплектических групп и в [75] для всех оставшихся групп лиева типа. Более того, в той же работе [75] авторами утверждалось, что последний оставшийся случай, когда 2, 3 ∈ π, а p / ∈ π, также полностью разобран и, тем самым, класси-фикация π-холловых подгрупп во всех конечных простых группах завершена.…”

Section: холловы подгруппы конечных простых группunclassified

“…Случай, когда 2 ∈ π, а 3, p / ∈ π, разобран в [45] для линейных и симплектических групп и в [75] для всех оставшихся групп лиева типа. Более того, в той же работе [75] авторами утверждалось, что последний оставшийся случай, когда 2, 3 ∈ π, а p / ∈ π, также полностью разобран и, тем самым, класси-фикация π-холловых подгрупп во всех конечных простых группах завершена. Но, к сожалению, этот результат базировался на неверной лемме 3.14 из [75].…”

Section: холловы подгруппы конечных простых группunclassified

“…Теорема 4.9 ([44], [75], mod CFSG). Предположим, что множество π про-стых чисел таково, что 2 / ∈ π или 3 / ∈ π. В этом случае конечная группа G обладает свойством E π в том и только том случае, когда каждый компози-ционный фактор группы G обладает этим свойством.…”

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Vdovin¹,

Vdovin²,

Ревин³

2011

Uspekhi Mat. Nauk

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The {ie210-01}-property in finite simple groups

Ревин

2008

Algebra Logic

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Let π be some set of primes. A finite group is said to possess the D π -property if all of its maximal π-subgroups are conjugate. It is not hard to show that this property is equivalent to satisfaction of the complete analog of Sylow's theorem for Hall π-subgroups of a group. In the paper, we bring to a close an arithmetic description of finite simple groups with the D π -property, for any set π of primes. Previously, it was proved that a finite group possesses the D π -property iff each composition factor of the group has this property. Therefore, the results obtained mean in fact that the question of whether a given group enjoys the D π -property becomes purely arithmetic.Hall π-subgroup, then G is said to possess the D π -property [1]. A group with the E π -(C π -, D π -) property is also referred to as an E π -(C π -, D π -) group, respectively.Hall subgroups and groups with the D π -property were studied by many well-known authors (see, e.g., ). A classical result in this direction is a celebrated theorem by Hall, which was proved in [2,3], and independently, in [4]. The Hall theorem, we recall, says that a finite group is soluble iff it has the D π -property, for any set π of primes.Let π be fixed. Then the class of groups with the D π -property may turn out to be substantially wider than the class of all soluble groups, since, say, every π-or π -group possesses this property. Thus, it seems natural to pose the following: Problem 1. For every set π, describe all groups with the D π -property. The present paper brings to a close a series of works [28][29][30][31][32][33] devoted to this problem and contains a result which solves it out.The Hall theorem demonstrates that there exists a close connection between the composition structure of a finite group and the existence of the D π -property in that group, for all π. It turns out that this connection obtains even if the set π is fixed. It is not hard to show that every factor group of a group with the D π -property also has this property. Below are some problems which have been intensively studied in the mathematical literature starting in the 1950s (see [1, 5-13, 19, 20, 22]). Problem 2. Will an extension of a groupA by a group B possess the D π -property if A and B have this property? Problem 3. Will a normal subgroup of a group with the D π -property possess this property? In the general setting, Problem 2 was formulated in [13; 34, Question 3.62], and Problem 3 in [22; 34, Question 13.33]. Ph. Hall brought out a positive solution to Problem 2 for the case where a Hall π-subgroup is nilpotent in A and is soluble in B (see [1, Thm. D5]). Shemetkov in [10] solved this problem for the case where Sylow p-subgroups of A are cyclic for all p ∈ π, and obtained a series of other important results in this direction (see [11,12]). In [19] and [20], first attempts were made to use the theory of simple groups in dealing with Problem 2. Although a positive solution was obtained for several particular cases, the general approach to Problems 2 and 3 was outlined only in ...

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Hall subgroups of finite groups

Cited by 36 publications

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An existence criterion for Hall subgroups of finite groups

An existence criterion for Hall subgroups of finite groups

Теоремы Силовского Типа

The {ie210-01}-property in finite simple groups

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