A finite group in which all non-nilpotent maximal subgroups are normal has a Sylow tower

Let G be a finite group. A subgroup H of G is called a 2-maximal subgroup of G if there exists a maximal subgroup M of G such that H is a maximal subgroup of M. In this paper, we discuss the influence of π-quasinormality of 2-maximal subgroups of Sylow subgroups on the structure of a finite group, and obtain some sufficient conditions under which the finite group is p-nilpotent, supersolvable, or possesses an ordered Sylow tower.

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ON A CLASS OF SUPERSOLUBLE GROUPS

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972714000306 ◽

2014 ◽

Vol 90 (2) ◽

pp. 220-226 ◽

Cited By ~ 3

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.

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A generalized Frattini subgroup of a finite group

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s016117128900030x ◽

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.

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

Algebra Colloquium ◽

10.1142/s1005386711000538 ◽

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.

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Finite groups with two supersoluble subgroups

Journal of Group Theory ◽

10.1515/jgth-2018-0150 ◽

2019 ◽

Vol 22 (2) ◽

pp. 297-312 ◽

Cited By ~ 6

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

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500023683 ◽

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.

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A Lower Bound on the Number of Maximal Subgroups in a Finite Group

Bulletin of the Iranian Mathematical Society ◽

10.1007/s41980-019-00345-w ◽

2020 ◽

Vol 46 (6) ◽

pp. 1599-1602

Author(s):

Jiakuan Lu ◽

Shenyang Wang ◽

Wei Meng

Keyword(s):

Finite Group ◽

Lower Bound ◽

Maximal Subgroups ◽

A Finite Group

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A polynomial bound for the number of maximal systems of imprimitivity of a finite transitive permutation group

Forum Mathematicum ◽

10.1515/forum-2019-0222 ◽

2020 ◽

Vol 32 (3) ◽

pp. 713-721

Author(s):

Andrea Lucchini ◽

Mariapia Moscatiello ◽

Pablo Spiga

Keyword(s):

Finite Group ◽

Permutation Group ◽

Maximal Subgroups ◽

Transitive Permutation Group ◽

Classic Result ◽

A Finite Group

AbstractWe show that there exists a constant a such that, for every subgroup H of a finite group G, the number of maximal subgroups of G containing H is bounded above by {a\lvert G:H\rvert^{3/2}}. In particular, a transitive permutation group of degree n has at most {an^{3/2}} maximal systems of imprimitivity. When G is soluble, generalizing a classic result of Tim Wall, we prove a much stronger bound, that is, the number of maximal subgroups of G containing H is at most {\lvert G:H\rvert-1}.

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