Normalizers Of Sylow Subgroups Research Articles

Correction to: Normalizers of Sylow Subgroups in Finite Linear and Unitary Groups

Correction to: Normalizers of Sylow Subgroups in Finite Linear and Unitary Groups

Normalizers of Sylow subgroups in finite linear and unitary groups

We specify normalizers of Sylow r-subgroups in finite simple linear and unitary groups for the case where r is an odd prime distinct from the characteristic of a definition field of a group.

Normalizers of Sylow Subgroups in Finite Linear and Unitary Groups

We specify normalizers of Sylow r-subgroups in finite simple linear and unitary groups for the case where r is an odd prime distinct from the characteristic of a definition field of a group.

Finite groups with certain normalizers of Sylow subgroups

In this paper, we prove that a finite group [Formula: see text] is nilpotent if and only if for every [Formula: see text] the normalizer of each Sylow [Formula: see text]-subgroup of [Formula: see text] is [Formula: see text]-nilpotent, where [Formula: see text] is a prime [Formula: see text] there exists some maximal subgroup [Formula: see text] of [Formula: see text] such that [Formula: see text] divides [Formula: see text].

A survey on Sylow normalizers and classes of groups

We report on a stream of research in relation with Sylow normalizers, i.e. normalizers of Sylow subgroups, of nite groups and group’s classes. Mathematics Subject Classication: 20D20, 20F17, 20D15

ON SYLOW NORMALIZERS OF FINITE GROUPS

The paper considers the influence of Sylow normalizers, i.e. normalizers of Sylow subgroups, on the structure of finite groups. In the universe of finite soluble groups it is known that classes of groups with nilpotent Hall subgroups for given sets of primes are exactly the subgroup-closed saturated formations satisfying the following property: a group belongs to the class if and only if its Sylow normalizers do so. The paper analyzes the extension of this research to the universe of all finite groups.

Solvability of any group with Hall supplements to normalizers of Sylow subgroups

The solvability of any finite group with Hall supplements to normalizers of the Sylow subgroups is established.

On the Influence of the Indices of Normalizers of Sylow Subgroups on the Structure of a Finite p-Soluble Group

We study the dependence of the structure of finite p-soluble groups on the indices of normalizers of Sylow subgroups. We obtain estimates for the p-length of these groups, and for small values of indices we find the nilpotent length of a soluble group.

On normalizers of Sylow subgroups in finite groups

It is a wen-known result of Glauberman [1] that the order of a finite group G is some power of a prime if for every prime q which .di.'vides the order of G the normalizer of each Sylow q-subgroup of G is a q-group. This result naturally suggests the question whether a finite group G is nilpotent if the normalizer of each Sylow p-subgroup of G is p-nilpotent for every prime p. The first attempt to solve this question is due to I. P. Doktorov (see [2]) who proved that a finite group G is nilpotent if the normalizer of every Sylow p-subgroup of G is a p-nilpotent Hall subgroup of G for every prime p. This question was also considered in [3] but this article has a gap. The purpose of the present article is to settle this question. All. groups considered in the present article are finite. Wherever possible we follow the notation and terminology of [4, 5]. We say that a set ~r of primes is nice if {2, 3} is not contained in a" and every prime in ~r \ {2, 3} is greater than every prime in ~r', where ~r' is the complement of ~r in the set ~v of aLl primes.

Finite groups with given indices of normalizers of Sylow subgroups

Finite groups with given indices of normalizers of Sylow subgroups

Finite Groups With Given Normalizers of Sylow Subgroups

A series of work was devoted to investigation of finite groups with the given properties of Sylow subgroups in the recent years.In Ref. [1] it was proved that if the normalizer of any non-unit Sylow subgroup of a finite group G is nilpotent, then

Structure of Sylow 2-subgroups of the alternating groups and normalizers of Sylow subgroups in the symmetric and alternating groups

Structure of Sylow 2-subgroups of the alternating groups and normalizers of Sylow subgroups in the symmetric and alternating groups

Finite groups with complemented normalizers of Sylow subgroups

Buchthal [3] considered a finite group G in which the normalizers of the nonidentity Sylow subgroups have Hall complements, and he proved that if these complements are Abelian, then G is solvable, and if they are cyclic, then G is supersolvable. In this present note, which is a continuation of [4], we generalize the results of [3] and show that the Abelian and cyclic properties can be weakened, respectively, to nilpotency and the property of being a Z-group; also, in the case where the normalizer of each Sylow subgroup of G has a nilpotent Hall complement, we will not only prove that G is solvable, but we will indicate its structure, which turns out to be close to that of nilpotent groups (Theorem 3). The proof of this theorem depends essentially on Theorem i, which gives a necessary and sufficient condition for a finite group to be nilpotent, a corollary of which is a result of Glauberman [5]. The results of this present note were announced in [6]. We will use the following definitions and notation: G is a finite group; IHI is the order of the subgroup H; ~(H) is the set of prime divisors of IHI; Nx(H) is the normalizer of the subgroup H in the group X ; the normalizer of a subgroup H in the group G will be denoted, for=6onvenience, by N(H) with no subscript; C(H) is the centralizer of the subgroup H in the group G; p, q, r, . . . are primes; Gp is a Sylow p'subgroup of G; GP is a Hall pcomplement of G; i.e., G = GpGP. A complement D of a subgroup H in G is called a Hall complement if D is a Hall subgroup of G, i.e., if the order and the index of D in G are coprime. A group is called p-decomposable if it decomposes into a direct product of a Sylow psubgroup and a Sylow p-complement [7].