scholarly journals Kernel systems on finite groups

2001 ◽  
Vol 163 ◽  
pp. 71-85 ◽  
Author(s):  
Paul Lescot
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Prime Factor ◽  
A Priori ◽  
Solvable Subgroup ◽  
Natural Conditions ◽  
Sylow Subgroups ◽  
Frobenius Kernel ◽  
Maximal Solvable Subgroup

We introduce a notion of kernel systems on finite groups: roughly speaking, a kernel system on the finite group G consists in the data of a pseudo-Frobenius kernel in each maximal solvable subgroup of G, subject to certain natural conditions. In particular, each finite CA-group can be equipped with a canonical kernel system. We succeed in determining all finite groups with kernel system that also possess a Hall p′-subgroup for some prime factor p of their order; this generalizes a previous result of ours (Communications in Algebra 18(3), 1990, pp. 833-838). Remarkable is the fact that we make no a priori abelianness hypothesis on the Sylow subgroups.

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 two supersoluble subgroups

2019 ◽  
Vol 22 (2) ◽  
pp. 297-312 ◽  
Author(s):  
Victor S. Monakhov ◽  
Alexander A. Trofimuk
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Sufficient Conditions ◽  
Maximal Subgroups ◽  
Sylow Subgroups ◽  
Derived Subgroup ◽  
Nilpotent Residual ◽  
A Finite Group

AbstractLetGbe a finite group. In this paper we obtain some sufficient conditions for the supersolubility ofGwith two supersoluble non-conjugate subgroupsHandKof prime index, not necessarily distinct. It is established that the supersoluble residual of such a group coincides with the nilpotent residual of the derived subgroup. We prove thatGis supersoluble in the following cases: one of the subgroupsHorKis nilpotent; the derived subgroup{G^{\prime}}ofGis nilpotent;{|G:H|=q>r=|G:K|}andHis normal inG. Also the supersolubility ofGwith two non-conjugate maximal subgroupsMandVis obtained in the following cases: all Sylow subgroups ofMand ofVare seminormal inG; all maximal subgroups ofMand ofVare seminormal inG.

scholarly journals Mutually Permutable Products of Finite Groups

ISRN Algebra ◽  
2011 ◽  
Vol 2011 ◽  
pp. 1-4
Author(s):  
Rola A. Hijazi
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Sylow Subgroups ◽  
A Finite Group ◽  
Mutually Permutable Products

Let G be a finite group and G1, G2 are two subgroups of G. We say that G1 and G2 are mutually permutable if G1 is permutable with every subgroup of G2 and G2 is permutable with every subgroup of G1. We prove that if is the product of three supersolvable subgroups G1, G2, and G3, where Gi and Gj are mutually permutable for all i and j with and the Sylow subgroups of G are abelian, then G is supersolvable. As a corollary of this result, we also prove that if G possesses three supersolvable subgroups whose indices are pairwise relatively prime, and Gi and Gj are mutually permutable for all i and j with , then G is supersolvable.

scholarly journals Finite Groups with Certain Permutability Criteria

2019 ◽  
Vol 12 (2) ◽  
pp. 571-576 ◽  
Author(s):  
Rola A. Hijazi ◽  
Fatme M. Charaf
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Sylow Subgroups ◽  
Group A ◽  
A Finite Group

Let G be a finite group. A subgroup H of G is said to be S-permutable in G if itpermutes with all Sylow subgroups of G. In this note we prove that if P, the Sylowp-subgroup of G (p > 2), has a subgroup D such that 1 <|D|<|P| and all subgroups H of P with |H| = |D| are S-permutable in G, then G′ is p-nilpotent.

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].

On SS-supplemented subgroups of some sylow subgroups of finite groups

2021 ◽  
Author(s):  
Xuanli He ◽  
Qinghong Guo ◽  
Muhong Huang
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Necessary Conditions ◽  
Saturated Formation ◽  
Sylow Subgroups ◽  
Group A ◽  
Sufficient And Necessary Conditions ◽  
A Finite Group

Let [Formula: see text] be a finite group. A subgroup [Formula: see text] of [Formula: see text] is called to be [Formula: see text]-permutable in [Formula: see text] if [Formula: see text] permutes with all Sylow subgroups of [Formula: see text]. A subgroup [Formula: see text] of [Formula: see text] is said to be [Formula: see text]-supplemented in [Formula: see text] if there exists a subgroup [Formula: see text] of [Formula: see text] such that [Formula: see text] and [Formula: see text] is [Formula: see text]-permutable in [Formula: see text]. In this paper, we investigate [Formula: see text]-nilpotency of a finite group. As applications, we give some sufficient and necessary conditions for a finite group belongs to a saturated formation.

Finite Groups with Four Relative Commutativity Degrees

2015 ◽  
Vol 22 (03) ◽  
pp. 449-458 ◽  
Author(s):  
A. Erfanian ◽  
M. Farrokhi D.G.
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Frobenius Group ◽  
Maximal Subgroups ◽  
Frobenius Kernel ◽  
A Finite Group

It is shown that a finite group G has four relative commutativity degrees if and only if G/Z(G) is a p-group of order p3 and G has no abelian maximal subgroups, or G/Z(G) is a Frobenius group with Frobenius kernel and complement isomorphic to ℤp × ℤp and ℤq, respectively, and the Sylow p-subgroup of G is abelian, where p and q are distinct primes.

Local systems and Sylow subgroups in locally finite groups. II

1974 ◽  
Vol 75 (1) ◽  
pp. 1-22 ◽  
Author(s):  
A. Rae
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Local System ◽  
The Other ◽  
Locally Finite Group ◽  
Local Systems ◽  
Locally Finite ◽  
Sylow Subgroups ◽  
Locally Finite Groups ◽  
Other Hand

1.1. Introduction. In this paper, we continue with the theme of (1): the relationships holding between the Sπ (i.e. maximal π) subgroups of a locally finite group and the various local systems of that group. In (1), we were mainly concerned with ‘good’ Sπ subgroups – those which reduce into some local system (and are said to be good with respect to that system). Here, on the other hand, we are concerned with a very much more special sort of Sπ subgroup.

New characterizations of p-soluble and p-supersoluble finite groups

2012 ◽  
Vol 49 (3) ◽  
pp. 390-405
Author(s):  
Wenbin Guo ◽  
Alexander Skiba
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Maximal Subgroup ◽  
Finite Groups ◽  
Prime Order ◽  
Cyclic Subgroup ◽  
Sylow Subgroups ◽  
Quasinormal Subgroup ◽  
A Finite Group

Let G be a finite group and H a subgroup of G. 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 and HsG the intersection of all S-quasinormal subgroups of G containing H. The symbol |G|p denotes the order of a Sylow p-subgroup of G. We prove the followingTheorem A. Let G be a finite group and p a prime dividing |G|. Then G is p-supersoluble if and only if for every cyclic subgroup H ofḠ (G) of prime order or order 4 (if p = 2), Ḡhas a normal subgroup T such thatHsḠandH∩T=HsḠ∩T.Theorem B. A soluble finite group G is p-supersoluble if and only if for every 2-maximal subgroup E of G such that Op′ (G) ≦ E and |G: E| is not a power of p, G has an S-quasinormal subgroup T with cyclic Sylow p-subgroups such that EsG = ET and |E ∩ T|p = |EsG ∩ T|p.Theorem C. A finite group G is p-soluble if for every 2-maximal subgroup E of G such that Op′ (G) ≦ E and |G: E| is not a power of p, G has an S-quasinormal subgroup T such that EsG = ET and |E ∩ Tp = |EsG ∩ T|p.

On p-supersolvability of finite groups

2015 ◽  
Vol 52 (4) ◽  
pp. 504-510
Author(s):  
Mohamed Asaad
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Normal Closure ◽  
Sylow Subgroups ◽  
Group A ◽  
A Finite Group

Let G be a finite group. A subgroup H of G is said to be s-permutable in G if H permutes with all Sylow subgroups of G. Let H be a subgroup of G and let HsG be the subgroup of H generated by all those subgroups of H which are s-permutable in G. A subgroup H of G is called n-embedded in G if G has a normal subgroup T such that HG = HT and H ∩ T ≦ HsG, where HG is the normal closure of H in G. We investigate the influence of n-embedded subgroups of the p-nilpotency and p-supersolvability of G.

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