scholarly journals Investigation of Some Subgroups in Determining the Frattini Subgroup of Non-Abelian Finite Groups

2021 ◽  
Vol 09 (10) ◽  
Author(s):  
Shuaibu Garba Ngulde ◽  
Keyword(s):  
Abelian Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Maximum Size ◽  
Maximal Subgroups ◽  
Fitting Subgroup ◽  
Frattini Subgroup ◽  
Dihedral Groups ◽  
Parent Group ◽  
Internal Properties

Frattini subgroup, Φ(G), of a group G is the intersection of all the maximal subgroups of G, or else G itself if G has no maximal subgroups. If G is a p-group, then Φ(G) is the smallest normal subgroup N such the quotient group G/N is an elementary abelian group. It is against this background that the concept of p-subgroup and fitting subgroup play a significant role in determining Frattini subgroup (especially its order) of dihedral groups. A lot of scholars have written on Frattini subgroup, but no substantial relationship has so far been identified between the parent group G and its Frattini subgroup Φ(G) which this tries to establish using the approach of Jelten B. Napthali who determined some internal properties of non abelian groups where the centre Z(G) takes its maximum size.

scholarly journals Maximal subgroups of finite groups

1990 ◽  
Vol 13 (2) ◽  
pp. 311-314
Author(s):  
S. Srinivasan
Keyword(s):  
Maximal Subgroup ◽  
Finite Groups ◽  
Maximal Subgroups ◽  
Fitting Subgroup ◽  
Small Index

In finite groups maximal subgroups play a very important role. Results in the literature show that if the maximal subgroup has a very small index in the whole group then it influences the structure of the group itself. In this paper we study the case when the index of the maximal subgroups of the groups have a special type of relation with the Fitting subgroup of the group.

scholarly journals A remark on the C-normality of maximal subgroups of finite groups

1997 ◽  
Vol 40 (2) ◽  
pp. 243-246
Author(s):  
Yanming Wang
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Maximal Subgroups ◽  
Primitive Groups ◽  
A Finite Group

A subgroup H is called c-normal in a group G if there exists a normal subgroup N of G such that HN = G and H∩N ≤ HG, where HG =: Core(H) = ∩g∈GHg is the maximal normal subgroup of G which is contained in H. We use a result on primitive groups and the c-normality of maximal subgroups of a finite group G to obtain results about the influence of the set of maximal subgroups on the structure 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 On Prefrattini residuals

1998 ◽  
Vol 40 (2) ◽  
pp. 187-197
Author(s):  
A. Ballester-Bolinches ◽  
H. Bechtell ◽  
L. M. Ezquerro
Keyword(s):  
Normal Subgroup ◽  
Maximal Subgroup ◽  
Finite Groups ◽  
Minimal Normal Subgroup ◽  
Classification Problem ◽  
Chief Factor ◽  
Maximal Subgroups ◽  
Normal Subgroups ◽  
Chief Factors ◽  
Will Force

All groups considered in the sequel are finite. Let (ℭ and denote the formations of groups which consist of collections of groups that respectively either split over each normal subgroup (nC-groups) or for which the groups do not possess nontrivial Frattini chief factors [8]. The purpose of this article is to develop and expand a concept that arises naturally with the residuals for these formations, namely each G-chief factor is non-complemented (Frattini). With respect to a solid set X of maximal subgroups, these properties are generalized respectively to so-called X-parafrattini (X-profrattini) normal subgroups for which each type is closed relative to products. The relationships among the unique maximal normal subgroups that result from these products, the solid set of maximal subgroups X, X-prefrattini subgroups, and the residuals of formations are explored. This leads to a well-defined collected of formations, the partially nonsaturated formations, with properties analogous to those which are totally non-saturated. In the development, attention is given to a set of maximal subgroups which is the image of a solid function defined on all groups, a weaker condition than that of a solid set. A result of particular interest answers affirmatively the long-standing conjecture that a non-trivial nC-group G is solvable if and only if each G-chief factor is complemented by a maximal subgroup. This will force a critical re-examination of the classification problem for nC-groups. Since the article continues the investigations on finite groups initiated in [2], a familiarity with that article is assumed. All other notation and terminology is from [6]. If M is a maximal subgroup of a group G and G/C or e G(M) is a monolithic primitive group, i.e. a group with a unique minimal normal subgroup, then M is called a monolithic maximal subgroupof G.

Generalized Frattini Subgroups of Finite Groups. II

1969 ◽  
Vol 21 ◽  
pp. 418-429 ◽  
Author(s):  
James C. Beidleman
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Proper Subgroup ◽  
Subnormal Subgroup ◽  
Normal Subgroups ◽  
Frattini Subgroup ◽  
A Finite Group ◽  
Proper Normal Subgroup ◽  
Equivalent Conditions

The theory of generalized Frattini subgroups of a finite group is continued in this paper. Several equivalent conditions are given for a proper normal subgroup H of a finite group G to be a generalized Frattini subgroup of G. One such condition on H is that K is nilpotent for each normal subgroup K of G such that K/H is nilpotent. From this result, it follows that the weakly hyper-central normal subgroups of a finite non-nilpotent group G are generalized Frattini subgroups of G.Let H be a generalized Frattini subgroup of G and let K be a subnormal subgroup of G which properly contains H. Then H is a generalized Frattini subgroup of K.Let ϕ(G) be the Frattini subgroup of G. Suppose that G/ϕ(G) is nonnilpotent, but every proper subgroup of G/ϕ(G) is nilpotent. Then ϕ(G) is the unique maximal generalized Frattini subgroup of G.

On metanilpotent groups satisfying the minimal condition on normal subgroups

2019 ◽  
Vol 22 (5) ◽  
pp. 809-836
Author(s):  
Derek J. S. Robinson
Keyword(s):  
Maximal Subgroups ◽  
Minimal Condition ◽  
Schur Multiplier ◽  
Normal Subgroups ◽  
Fitting Subgroup ◽  
Frattini Subgroup ◽  
Comprehensive Account

Abstract A comprehensive account is given of the theory of metanilpotent groups with the minimal condition on normal subgroups. After reviewing classical material, many new results are established relating to the Fitting subgroup, the Hirsch–Plotkin radical, the Frattini subgroup, splitting and conjugacy, the Schur multiplier, Sylow structure and the maximal subgroups. Module theoretic and homological methods are used throughout.

A generalized Frattini subgroup

2020 ◽  
pp. 2150026
Author(s):  
Ruslan V. Borodich
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Subnormal Subgroup ◽  
Maximal Subgroups ◽  
General Question ◽  
Frattini Subgroup ◽  
Local Formation ◽  
Glasgow Math ◽  
A Finite Group ◽  
Subnormal Subgroups

In the work of Beidleman and Smith [On Frattini-like subgroups, Glasgow Math. J. 35 (1993) 95–98], the following question was raised: “If [Formula: see text] is a subnormal subgroup of a finite group [Formula: see text] containing [Formula: see text], then whether the supersolvability of [Formula: see text] follows the supersolvability of [Formula: see text]”. This problem was considered in works of Selkin [Maximal Subgroups in the Theory of Classes of Finite Groups (Belaruskaya, Navuka, 1997)], Skiba [On the intersection of all maximal [Formula: see text]-subgroups of a finite group, Prob. Phys. Math. Tech. 3(4) (2010) 56–62], Ballester-Bolinches [On [Formula: see text]-subnormal subgroups and Frattini-like subgroups of a finite group, Glasgow Math. J. 36 (1994) 241–247] and many other authors (see monograph [Maximal Subgroups in the Theory of Classes of Finite Groups (Belaruskaya, Navuka, 1997)]). In this paper, we give the answer to the more general question: “Let [Formula: see text] be a local formation. If [Formula: see text] is a subnormal subgroup of a group [Formula: see text], then in what case [Formula: see text] will follow from [Formula: see text]”.

A Frattini-like subgroup

1975 ◽  
Vol 77 (2) ◽  
pp. 247-257 ◽  
Author(s):  
M. J. Tomkinson
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Maximal Subgroups ◽  
Frattini Subgroup ◽  
A Finite Group

The Frattini subgroup φ(G) of a group G is the intersection of G and all its maximal subgroups. The following results for finite groups are well known:THEOREM A0. If G is a finite group, then the following three conditions are equivalent:(i) G is nilpotent,(ii) G/φ(G) is nilpotent,(iii) φ(G) ≥ G′.

ON A THEOREM OF HUPPERT

2011 ◽  
Vol 10 (02) ◽  
pp. 295-301
Author(s):  
JIANGTAO SHI ◽  
CUI ZHANG
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Soluble Group ◽  
Proper Subgroup ◽  
Maximal Subgroups ◽  
Frattini Subgroup ◽  
A Finite Group ◽  
Group 1

A well-known theorem of Huppert states that a finite group is soluble if its every proper subgroup is supersoluble. In this paper, we proved the following result: let G be a finite group. (1) If G has exactly n non-supersoluble proper subgroups, where 0 ≤ n ≤ 7 and n ≠ 5, then G is soluble. (2) G is a non-soluble group with exactly five non-supersoluble proper subgroups if and only if all non-supersoluble proper subgroups are conjugate maximal subgroups and G/Φ(G) ≅ A5, where Φ(G) is the Frattini subgroup of G. Furthermore, we also considered the influence of the number of non-abelian proper subgroups on the solubility of finite groups.

On weakly ℨ-permutable subgroups of finite groups

2015 ◽  
Vol 14 (05) ◽  
pp. 1550062 ◽  
Author(s):  
A. A. Heliel ◽  
M. M. Al-Shomrani ◽  
T. M. Al-Gafri
Keyword(s):  
Finite Groups ◽  
Subnormal Subgroup ◽  
Maximal Subgroups ◽  
Supersolvable Groups ◽  
Fitting Subgroup ◽  
Permutable Subgroup ◽  
Sylow Subgroups ◽  
Permutable Subgroups ◽  
Complete Set ◽  
A Finite Group

Let ℨ be a complete set of Sylow subgroups of a finite group G, that is, for each prime p dividing the order of G, ℨ contains exactly one and only one Sylow p-subgroup of G. A subgroup H of G is said to be ℨ-permutable of G if H permutes with every member of ℨ. A subgroup H of G is said to be a weakly ℨ-permutable subgroup of G if there exists a subnormal subgroup K of G such that G = HK and H ∩ K ≤ Hℨ, where Hℨ is the subgroup of H generated by all those subgroups of H which are ℨ-permutable subgroups of G. In this paper, we prove that if p is the smallest prime dividing the order of G and the maximal subgroups of Gp ∈ ℨ are weakly ℨ-permutable subgroups of G, then G is p-nilpotent. Moreover, we prove that if 𝔉 is a saturated formation containing the class of all supersolvable groups, then G ∈ 𝔉 iff there is a solvable normal subgroup H in G such that G/H ∈ 𝔉 and the maximal subgroups of the Sylow subgroups of the Fitting subgroup F(H) are weakly ℨ-permutable subgroups of G. These two results generalize and unify several results in the literature.

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