The structure of an infinite Sylow subgroup in some periodic Shunkov groups

2002 ◽  
Vol 12 (5) ◽  
Author(s):  
V.I. Senashov
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Sylow Subgroup ◽  
Group Influence ◽  
Burnside Groups ◽  
Strongly Embedded Subgroup ◽  
Periodic Groups ◽  
Finite Subgroups ◽  
Trivial Subgroup

AbstractWe study periodic groups such that the normaliser of any finite non-trivial subgroup of such a group is almost layer-finite. The class of groups satisfying this condition is rather wide and includes the free Burnside groups of odd period which is greater than 665 and the groups constructed by A. Yu. Olshanskii.We consider the classical question: how the properties of the system of subgroups of a group influence on the properties of the group? We show that almost layer-finiteness is transferred on the group G from the normalisers of non-trivial finite subgroups of the group G if G is periodic conjugately biprimitively finite group with a strongly embedded subgroup.We study the structure of an infinite Sylow 2-subgroup in a periodic conjugately biprimitively finite group in the case that the normaliser of any finite non-trivial subgroup is almost layer-finite.The results of the paper can be useful in the study of the class of periodic conjugately biprimitively finite groups (periodic Shunkov groups).

Finite Groups with Some Subgroups of Sylow Subgroups s∗-Semipermutable

2021 ◽  
Vol 58 (2) ◽  
pp. 147-156
Author(s):  
Qingjun Kong ◽  
Xiuyun Guo
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Sylow Subgroup ◽  
Subnormal Subgroup ◽  
Sylow Subgroups ◽  
A Finite Group

We introduce a new subgroup embedding property in a finite group called s∗-semipermutability. Suppose that G is a finite group and H is a subgroup of G. H is said to be s∗-semipermutable in G if there exists a subnormal subgroup K of G such that G = HK and H ∩ K is s-semipermutable in G. We fix in every non-cyclic Sylow subgroup P of G some subgroup D satisfying 1 < |D| < |P | and study the structure of G under the assumption that every subgroup H of P with |H | = |D| is s∗-semipermutable in G. Some recent results are generalized and unified.

FINITE GROUPS WITH GIVEN NEARLY S-QUASINORMAL SUBGROUPS

2008 ◽  
Vol 01 (03) ◽  
pp. 369-382
Author(s):  
Nataliya V. Hutsko ◽  
Vladimir O. Lukyanenko ◽  
Alexander N. Skiba
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Sylow Subgroup ◽  
Cyclic Subgroup ◽  
Saturated Formation ◽  
Supersoluble Groups ◽  
Sylow Subgroups ◽  
Quasinormal Subgroup ◽  
A Finite Group

Let G be a finite group and H a subgroup of G. Then H is said to be S-quasinormal in G if HP = PH for all Sylow subgroups P of G. Let HsG be the subgroup of H generated by all those subgroups of H which are S-quasinormal in G. Then we say that H is nearly S-quasinormal in G if G has an S-quasinormal subgroup T such that HT = G and T ∩ H ≤ HsG. Our main result here is the following theorem. Let [Formula: see text] be a saturated formation containing all supersoluble groups and G a group with a normal subgroup E such that [Formula: see text]. Suppose that every non-cyclic Sylow subgroup P of E has a subgroup D such that 1 < |D| < |P| and all subgroups H of P with order |H| = |D| and every cyclic subgroup of P with order 4 (if |D| = 2 and P is a non-abelian 2-group) having no supersoluble supplement in G are nearly S-quasinormal in G. Then [Formula: see text].

Finite Groups with Normal Normalizers

1968 ◽  
Vol 20 ◽  
pp. 1256-1260 ◽  
Author(s):  
C. Hobby
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Sylow Subgroup ◽  
A Finite Group

We say that a finite group G has property N if the normalizer of every subgroup of G is normal in G. Such groups are nilpotent since every Sylow subgroup is normal (the normalizer of a Sylow subgroup is its own normalizer). Thus it is sufficient to study p-groups which have property N. Note that property N is inherited by subgroups and factor groups.

scholarly journals Finite Groups with Minimal CSS-subgroups

2021 ◽  
Vol 14 (3) ◽  
pp. 1002-1014
Author(s):  
A. A. Heliel ◽  
R. A. Hijazi ◽  
S. M. Al-Shammari
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Sylow Subgroup ◽  
Group A ◽  
A Finite Group

Let G be a finite group. A subgroup H of G is called SS-quasinormal in G if there is a supplement B of H to G such that H permutes with every Sylow subgroup of B. A subgroup H of G is called CSS-subgroup in G if there exists a normal subgroup K of G such that G = HK and H ∩K is SS-quasinormal in G. In this paper, we investigate the influence of minimal CSS-subgroups of G on its structure. Our results improve and generalize several recent results in the literature.

scholarly journals On Periodic Groups Saturated with Finite Frobenius Groups

2021 ◽  
Vol 35 ◽  
pp. 73-86
Author(s):  
B. E. Durakov ◽  
◽  
A. I. Sozutov ◽  
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Frobenius Group ◽  
Finite Subgroup ◽  
Frobenius Groups ◽  
Locally Finite ◽  
Periodic Groups ◽  
Group A ◽  
Quotient Groups ◽  
Combinatorial Group

A group is called weakly conjugate biprimitively finite if each its element of prime order generates a finite subgroup with any of its conjugate elements. A binary finite group is a periodic group in which any two elements generate a finite subgroup. If $\mathfrak{X}$ is some set of finite groups, then the group $G$ saturated with groups from the set $\mathfrak{X}$ if any finite subgroup of $G$ is contained in a subgroup of $G$, isomorphic to some group from $\mathfrak{X}$. A group $G = F \leftthreetimes H$ is a Frobenius group with kernel $F$ and a complement $H$ if $H \cap H^f = 1$ for all $f \in F^{\#}$ and each element from $G \setminus F$ belongs to a one conjugated to $H$ subgroup of $G$. In the paper we prove that a saturated with finite Frobenius groups periodic weakly conjugate biprimitive finite group with a nontrivial locally finite radical is a Frobenius group. A number of properties of such groups and their quotient groups by a locally finite radical are found. A similar result was obtained for binary finite groups with the indicated conditions. Examples of periodic non locally finite groups with the properties above are given, and a number of questions on combinatorial group theory are raised.

Finite Groups with Certain S-Permutable and GS-Maximal Subgroups

2020 ◽  
Vol 27 (04) ◽  
pp. 661-668
Author(s):  
A.M. Elkholy ◽  
M.H. Abd-Ellatif
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Sylow Subgroup ◽  
Prime Order ◽  
Maximal Subgroups ◽  
A Finite Group

Let G be a finite group and H a subgroup of G. We say that H is S-permutable in G if H permutes with every Sylow subgroup of G. A group G is called a generalized smooth group (GS-group) if [G/L] is totally smooth for every subgroup L of G of prime order. In this paper, we investigate the structure of G under the assumption that each subgroup of prime order is S-permutable if the maximal subgroups of G are GS-groups.

scholarly journals Some New Applications of Weakly H-Embedded Subgroups of Finite Groups

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 158
Author(s):  
Li Zhang ◽  
Li-Jun Huo ◽  
Jia-Bao Liu
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Sylow Subgroup ◽  
Normal Closure ◽  
A Finite Group ◽  
New Applications ◽  
New Criterion

A subgroup H of a finite group G is said to be weakly H -embedded in G if there exists a normal subgroup T of G such that H G = H T and H ∩ T ∈ H ( G ) , where H G is the normal closure of H in G, and H ( G ) is the set of all H -subgroups of G. In the recent research, Asaad, Ramadan and Heliel gave new characterization of p-nilpotent: Let p be the smallest prime dividing | G | , and P a non-cyclic Sylow p-subgroup of G. Then G is p-nilpotent if and only if there exists a p-power d with 1 < d < | P | such that all subgroups of P of order d and p d are weakly H -embedded in G. As new applications of weakly H -embedded subgroups, in this paper, (1) we generalize this result for general prime p and get a new criterion for p-supersolubility; (2) adding the condition “ N G ( P ) is p-nilpotent”, here N G ( P ) = { g ∈ G | P g = P } is the normalizer of P in G, we obtain p-nilpotence for general prime p. Moreover, our tool is the weakly H -embedded subgroup. However, instead of the normality of H G = H T , we just need H T is S-quasinormal in G, which means that H T permutes with every Sylow subgroup of G.

scholarly journals A solvability condition for finite groups with nilpotent maximal subgroups

1970 ◽  
Vol 3 (2) ◽  
pp. 273-276
Author(s):  
John Randolph
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Maximal Subgroup ◽  
Finite Groups ◽  
Sylow Subgroup ◽  
Solvability Condition ◽  
Maximal Subgroups ◽  
A Finite Group ◽  
Nilpotent Maximal Subgroup

Let G be a finite group with a nilpotent maximal subgroup S and let P denote the 2-Sylow subgroup of S. It is shown that if P ∩ Q is a normal subgroup of P for any 2-Sylow subgroup Q of G, then G is solvable.

scholarly journals Finite Groups with Some -Supplemented Subgroups

Algebra ◽  
2013 ◽  
Vol 2013 ◽  
pp. 1-5
Author(s):  
Guo Zhong ◽  
Liying Yang ◽  
Huaquan Wei ◽  
Xuanlong Ma ◽  
Jiayi Xia
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Sylow Subgroup ◽  
Maximal Subgroups ◽  
Group A ◽  
A Finite Group

Let be a subgroup of a finite group , a prime dividing the order of , and a Sylow -subgroup of for prime We say that is -supplemented in if there is a subgroup of such that and where denotes the subgroup of generated by all those subgroups of which are -quasinormally embedded in In this paper, we characterize -nilpotency and supersolvability of under the assumption that all maximal subgroups of are -supplemented in .

scholarly journals Finite Groups Whose Certain Subgroups of Prime Power Order Are -Semipermutable

ISRN Algebra ◽  
2011 ◽  
Vol 2011 ◽  
pp. 1-8
Author(s):  
Mustafa Obaid
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Sylow Subgroup ◽  
Prime Power ◽  
Prime Power Order ◽  
Group A ◽  
Abelian Subgroups ◽  
A Finite Group

Let be a finite group. A subgroup of is said to be S-semipermutable in if permutes with every Sylow -subgroup of with . In this paper, we study the influence of S-permutability property of certain abelian subgroups of prime power order of a finite group on its structure.

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