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.

scholarly journals On Frattini-like subgroups

1993 ◽  
Vol 35 (1) ◽  
pp. 95-98 ◽  
Author(s):  
James C. Beidleman ◽  
Howard Smith
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Maximal Subgroups ◽  
Property A ◽  
Arbitrary Group ◽  
Close Relationship ◽  
Similar Vein ◽  
Definition Of ◽  
A Finite Group

For any group G, denote by φf(G respectively L(G)) the intersection of all maximal subgroups of finite index (respectively finite nonprime index) in G, with the usual provision that the subgroup concerned equals Gif no such maximals exist. The subgroup φf(G) was discussed in [1] in connection with a property v possessed by certain groups: a group G has v if and only if every nonnilpotent, normal subgroup of G has a finite, nonnilpotent G-image. It was shown there, for instance, that G/φf(G) has v for all groups G. The subgroup L(G), in the case where G is finite, was investigated at some length in [3], one of the main results being that L(G) is supersoluble. (A published proof of this result appears as Theorem 3 of [4]). The present paper is concerned with the role of L(G) in groups G having property v or a related property a, the definition of which is obtained by replacing “nonnilpotent” by “nonsupersoluble” in the definition of v. We also present a result (namely Theorem 4) which displays a close relationship between the subgroups L(G) and φf(G) in an arbitrary group G. Some of the results for finite groups in [3] are found to hold with rather weaker hypotheses and, in fact, remain true for groups with v or a. We recall that if a group has a it also has v ([2]Theorem 2) but not conversely. For example, G = (x, y: y-1xy = x2)has v but not a. It is a well-known result of Gaschütz ([8], 5.2.15) that, in a finite group G, if His a normal subgroup containing φ(G) such that H/φ(G) is nilpotent than His nilpotent. This remains true in the case where G is any group with v [1, Proposition 1]. Our first result is in a similar vein and is a generalization of Theorem 9 of [7] and Theorem 1.2.9 of [3], the latter of which states that, for a finite group G, if G/L(G) is supersoluble, then so is G.

Notes on Splitting Extensions of Groups

1968 ◽  
Vol 11 (3) ◽  
pp. 371-374 ◽  
Author(s):  
C.Y. Tang
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Commutator Subgroup ◽  
Similar Type ◽  
Frattini Subgroup ◽  
Abelian Normal Subgroup ◽  
A Finite Group

In [1] Gaschütz has shown that a finite group G splits over an abelian normal subgroup N if its Frattini subgroup ϕ(G) intersects N trivially. When N is a non-abelian nilpotent normal subgroup of G the condition ϕ(G)∩ N = 1 cannot be satisfied: for if N is non-abelian then the commutator subgroup C(N) of N is non-trivial. Now N is nilpotent, whence 1 ≠ C(N)⊂ϕ(N). Since G is a finite group, therefore, by (3, theorem 7.3.17) ϕ⊂ϕ(G). It follows that ϕ(G) ∩ N ≠ 1. Thus the condition ϕ(G) ∩ N = 1 must be modified. In §1 we shall derive some similar type of conditions for G to split over N when the restriction of N being an abelian normal subgroup is removed. In § 2 we shall give a characterization of splitting extensions of N in which every subgroup splits over its intersection with N.

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 the Intersection of a Class of Maximal Subgroups of a Finite Group

1987 ◽  
Vol 39 (3) ◽  
pp. 603-611 ◽  
Author(s):  
N. P. Mukherjee ◽  
Prabir Bhattacharya
Keyword(s):  
Finite Group ◽  
Maximal Subgroups ◽  
Frattini Subgroup ◽  
The Family ◽  
A Finite Group

Of late there has been considerable interest in the study of analogs of the Frattini subgroup of a finite group and the investigation of their properties, particularly their influence on the structure of the group, see [2-11], [14-16] and [18]. Gaschütz [11] and more recently Bechtell [2] and Rose [18] have considered extensively the intersection of the family of all non-normal, maximal subgroups of a finite group. Deskins [8] has discussed the intersection of the family of all maximal subgroups of a finite group whose indices are not divisible by a given prime. Bhatia [7] considered the intersection of the class of all maximal subgroups of a given group whose indices are composites. In this paper we investigate the intersection of another class of maximal subgroups and its relationship with the structure of the group. The subgroup we consider here contains the Frattini subgroup and also the two subgroups introduced in [8] and [7].

On the Intersection of a Family of Maximal Subgroups Containing the Sylow Subgroups of a Finite Group

1988 ◽  
Vol 40 (2) ◽  
pp. 352-359 ◽  
Author(s):  
N. P. Mukherjee ◽  
Prabir Bhattacharya
Keyword(s):  
Finite Group ◽  
Maximal Subgroups ◽  
Frattini Subgroup ◽  
Sylow Subgroups ◽  
The Family ◽  
A Finite Group ◽  
Main Definition

Given a finite group G, the Frattini subgroup of G, Φ(G) is defined to be the intersection of all the maximal subgroups of G. Of late there have been several attempts to consider generalizations of Φ(G). For example, Gaschutz [7] and Rose [13] have investigated the intersection of all non-normal, maximal subgroups of a finite group. Deskins [6] has discussed the intersection of the family of maximal subgroups of a finite group whose indices are co-prime to a given prime. In [4-5, 12] we have considered the investigation of the family of all maximal subgroups of a finite group whose indices are composite and co-prime to a given prime. We have obtained several results about the family . In this paper which is a sequel to [4] we prove some further results about this family indicating the interesting role it plays especially when G is solvable or p-solvable. First we recall the main definition from [4].

scholarly journals ON A CLASS OF SUPERSOLUBLE GROUPS

2014 ◽  
Vol 90 (2) ◽  
pp. 220-226 ◽  
Author(s):  
A. BALLESTER-BOLINCHES ◽  
J. C. BEIDLEMAN ◽  
R. ESTEBAN-ROMERO ◽  
M. F. RAGLAND
Keyword(s):  
Finite Group ◽  
Maximal Subgroups ◽  
Supersoluble Groups ◽  
Sylow Subgroups ◽  
A Finite Group

AbstractA subgroup $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}H$ of a finite group $G$ is said to be S-semipermutable in $G$ if $H$ permutes with every Sylow $q$-subgroup of $G$ for all primes $q$ not dividing $|H |$. A finite group $G$ is an MS-group if the maximal subgroups of all the Sylow subgroups of $G$ are S-semipermutable in $G$. The aim of the present paper is to characterise the finite MS-groups.

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.

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