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.

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 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.

scholarly journals Finite Groups with All Maximal Subgroups of Prime or Prime Square Index

1964 ◽  
Vol 16 ◽  
pp. 435-442 ◽  
Author(s):  
Joseph Kohler
Keyword(s):  
Finite Group ◽  
Maximal Subgroup ◽  
Finite Groups ◽  
Prime Order ◽  
Maximal Subgroups ◽  
Odd Order ◽  
Order Case ◽  
Chief Factors ◽  
Even Order ◽  
A Finite Group

In this paper finite groups with the property M, that every maximal subgroup has prime or prime square index, are investigated. A short but ingenious argument was given by P. Hall which showed that such groups are solvable.B. Huppert showed that a finite group with the property M, that every maximal subgroup has prime index, is supersolvable, i.e. the chief factors are of prime order. We prove here, as a corollary of a more precise result, that if G has property M and is of odd order, then the chief factors of G are of prime or prime square order. The even-order case is different. For every odd prime p and positive integer m we shall construct a group of order 2apb with property M which has a chief factor of order larger than m.

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.

Some sufficient conditions on solvability of finite groups

2016 ◽  
Vol 15 (03) ◽  
pp. 1650057 ◽  
Author(s):  
Wei Meng ◽  
Jiakuan Lu ◽  
Li Ma ◽  
Wanqing Ma
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Sufficient Conditions ◽  
Conjugacy Classes ◽  
Maximal Subgroups ◽  
Prime Divisors ◽  
Abelian Subgroups ◽  
A Finite Group

For a finite group [Formula: see text], the symbol [Formula: see text] denotes the set of the prime divisors of [Formula: see text] denotes the number of conjugacy classes of maximal subgroups of [Formula: see text]. Let [Formula: see text] denote the number of conjugacy classes of non-abelian subgroups of [Formula: see text] and [Formula: see text] denote the number of conjugacy classes of all non-normal non-abelian subgroups of [Formula: see text]. In this paper, we consider the finite groups with [Formula: see text] or [Formula: see text]. We show these groups are solvable.

scholarly journals Finite Groups with Some -Supplemented Subgroups

Algebra ◽  
2013 ◽  
Vol 2013 ◽  
pp. 1-5
Author(s):  
Guo Zhong ◽  
Liying Yang ◽  
Huaquan Wei ◽  
Xuanlong Ma ◽  
Jiayi Xia
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Sylow Subgroup ◽  
Maximal Subgroups ◽  
Group A ◽  
A Finite Group

Let be a subgroup of a finite group , a prime dividing the order of , and a Sylow -subgroup of for prime We say that is -supplemented in if there is a subgroup of such that and where denotes the subgroup of generated by all those subgroups of which are -quasinormally embedded in In this paper, we characterize -nilpotency and supersolvability of under the assumption that all maximal subgroups of are -supplemented in .

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