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′.

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]”.

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.

scholarly journals A generalized Frattini subgroup of a finite group

1989 ◽  
Vol 12 (2) ◽  
pp. 263-266
Author(s):  
Prabir Bhattacharya ◽  
N. P. Mukherjee
Keyword(s):  
Finite Group ◽  
Maximal Subgroups ◽  
Frattini Subgroup ◽  
Definition Of ◽  
A Finite Group

For a finite group G and an arbitrary prime p, letSP(G)denote the intersection of all maximal subgroups M of G such that [G:M] is both composite and not divisible by p; if no such M exists we setSP(G)= G. Some properties of G are considered involvingSP(G). In particular, we obtain a characterization of G when each M in the definition ofSP(G)is nilpotent.

On S-Quasinormal and C-Normal Subgroups of Prime Power Order in Finite Groups

2011 ◽  
Vol 18 (04) ◽  
pp. 685-692
Author(s):  
Xuanli He ◽  
Shirong Li ◽  
Xiaochun Liu
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Prime Power ◽  
Maximal Subgroups ◽  
Prime Power Order ◽  
Normal Subgroups ◽  
A Finite Group

Let G be a finite group, p the smallest prime dividing the order of G, and P a Sylow p-subgroup of G with the smallest generator number d. Consider a set [Formula: see text] of maximal subgroups of P such that [Formula: see text]. It is shown that if every member [Formula: see text] of is either S-quasinormally embedded or C-normal in G, then G is p-nilpotent. As its applications, some further results are obtained.

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 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.

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.

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 ON A GRAPH RELATED TO THE MAXIMAL SUBGROUPS OF A GROUP

2010 ◽  
Vol 81 (2) ◽  
pp. 317-328 ◽  
Author(s):  
MARCEL HERZOG ◽  
PATRIZIA LONGOBARDI ◽  
MERCEDE MAJ
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Connected Graph ◽  
Solvable Groups ◽  
Maximal Subgroups ◽  
Finite Solvable Group ◽  
Finitely Generated ◽  
Infinite Case ◽  
Locally Graded Group ◽  
A Finite Group

AbstractLet G be a finitely generated group. We investigate the graph ΓM(G), whose vertices are the maximal subgroups of G and where two vertices M1 and M2 are joined by an edge whenever M1∩M2≠1. We show that if G is a finite simple group then the graph ΓM(G) is connected and its diameter is 62 at most. We also show that if G is a finite group, then ΓM(G) either is connected or has at least two vertices and no edges. Finite groups G with a nonconnected graph ΓM(G) are classified. They are all solvable groups, and if G is a finite solvable group with a connected graph ΓM(G), then the diameter of ΓM(G) is at most 2. In the infinite case, we determine the structure of finitely generated infinite nonsimple groups G with a nonconnected graph ΓM(G). In particular, we show that if G is a finitely generated locally graded group with a nonconnected graph ΓM(G), then G must be finite.

Influence of the Strong Theta Completions on Finite Groups

2010 ◽  
Vol 17 (01) ◽  
pp. 59-64 ◽  
Author(s):  
Ni Du ◽  
Shirong Li
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Maximal Subgroups ◽  
A Finite Group

Let G be a finite group. By using the concept of strong θ-completions for maximal subgroups of G, we obtain in this paper some new conditions under which G is solvable or supersolvable.

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