Sylow towers in groups where the index of every non-nilpotent maximal subgroup is prime

2018 ◽  
Vol 17 (07) ◽  
pp. 1850119
Author(s):  
Jiangtao Shi
Keyword(s):  
Finite Group ◽  
Maximal Subgroup ◽  
A Finite Group ◽  
Sylow Tower ◽  
Nilpotent Maximal Subgroup

In this paper, we prove that if every non-nilpotent maximal subgroup of a finite group [Formula: see text] has prime index then [Formula: see text] has a Sylow tower.

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 Finite groups with a nilpotent maximal subgroup

1964 ◽  
Vol 4 (4) ◽  
pp. 449-451 ◽  
Author(s):  
Zvonimir Janko
Keyword(s):  
Finite Group ◽  
Maximal Subgroup ◽  
Finite Groups ◽  
A Finite Group ◽  
Nilpotent Maximal Subgroup

Let G be a finite group all of whose proper subgroups are nilpotent. Then by a theorem of Schmidt-Iwasawa the group G is soluble. But what can we say about a finite group G is only one maximal subgroup is nilpotent? Let G be a finite group with a nilpotent maximal subgroup M.

Finite groups with given non-nilpotent maximal subgroups of prime index

2019 ◽  
Vol 18 (05) ◽  
pp. 1950087
Author(s):  
Xiaolan Yi ◽  
Shiyang Jiang ◽  
S. F. Kamornikov
Keyword(s):  
Finite Group ◽  
Maximal Subgroup ◽  
Finite Groups ◽  
Maximal Subgroups ◽  
Subgroup Structure ◽  
A Finite Group ◽  
Nilpotent Maximal Subgroup

The subgroup structure of a finite group, under the assumption that its every non-nilpotent maximal subgroup has prime index, is studied in the paper.

An Extension of a Theorem of Janko on Finite Groups with Nilpotent Maximal Subgroups

1971 ◽  
Vol 23 (3) ◽  
pp. 550-552
Author(s):  
John W. Randolph
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Maximal Subgroup ◽  
Finite Groups ◽  
Maximal Subgroups ◽  
A Finite Group ◽  
Nilpotent Maximal Subgroup

Throughout this paper G will denote a finite group containing a nilpotent maximal subgroup S and P will denote the Sylow 2-subgroup of S. The largest subgroup of S normal in G will be designated by core (S) and the largest solvable normal subgroup of G by rad(G). All other notation is standard.Thompson [6] has shown that if P = 1 then G is solvable. Janko [3] then observed that G is solvable if P is abelian, a condition subsequently weakened by him [4] to the assumption that the class of P is ≦ 2 . Our purpose is to demonstrate the sufficiency of a still weaker assumption about P.

On the π-Quasinormality of 2-Maximal Subgroups of Sylow Subgroups of a Finite Group

2012 ◽  
Vol 19 (04) ◽  
pp. 657-664
Author(s):  
Songliang Chen ◽  
Yun Fan
Keyword(s):  
Finite Group ◽  
Maximal Subgroup ◽  
Sufficient Conditions ◽  
Maximal Subgroups ◽  
Sylow Subgroups ◽  
Group A ◽  
A Finite Group ◽  
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.

𝒫-characters and the structure of finite solvable groups

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Jiakuan Lu ◽  
Kaisun Wu ◽  
Wei Meng
Keyword(s):  
Finite Group ◽  
Maximal Subgroup ◽  
Solvable Group ◽  
Irreducible Character ◽  
Solvable Groups ◽  
Irreducible Constituent ◽  
A Finite Group

AbstractLet 𝐺 be a finite group. An irreducible character of 𝐺 is called a 𝒫-character if it is an irreducible constituent of (1_{H})^{G} for some maximal subgroup 𝐻 of 𝐺. In this paper, we obtain some conditions for a solvable group 𝐺 to be 𝑝-nilpotent or 𝑝-closed in terms of 𝒫-characters.

scholarly journals Extensions of finite nilpotent groups

1970 ◽  
Vol 2 (2) ◽  
pp. 267-274
Author(s):  
John Poland
Keyword(s):  
Finite Group ◽  
Maximal Subgroup ◽  
Solvable Groups ◽  
Nilpotent Groups ◽  
Theoretic Property ◽  
Joint Paper ◽  
The Core ◽  
A Finite Group

If G is a finite group and P is a group-theoretic property, G will be called P-max-core if for every maximal subgroup M of G, M/MG has property P where MG = ∩ is the core of M in G. In a joint paper with John D. Dixon and A.H. Rhemtulla, we showed that if p is an odd prime and G is (p-nilpotent)-max-core, then G is p-solvable, and then using the techniques of the theory of solvable groups, we characterized nilpotent-max-core groups as finite nilpotent-by-nilpotent groups. The proof of the first result used John G. Thompson's p-nilpotency criterion and hence required p > 2. In this paper I show that supersolvable-max-core groups (and hence (2-nilpotent)-max-core groups) need not be 2-solvable (that is, solvable). Also I generalize the second result, among others, and characterize (p-nilpotent)-max-core groups (for p an odd prime) as finite nilpotent-by-(p-nilpotent) groups.

scholarly journals ON WEAKLY -SUPPLEMENTED SUBGROUPS OF SYLOW p-SUBGROUPS OF FINITE GROUPS

2011 ◽  
Vol 53 (2) ◽  
pp. 401-410 ◽  
Author(s):  
LONG MIAO
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Maximal Subgroup ◽  
Finite Groups ◽  
Prime Divisor ◽  
Proper Subgroup ◽  
A Finite Group

AbstractA subgroup H is called weakly -supplemented in a finite group G if there exists a subgroup B of G provided that (1) G = HB, and (2) if H1/HG is a maximal subgroup of H/HG, then H1B = BH1 < G, where HG is the largest normal subgroup of G contained in H. In this paper we will prove the following: Let G be a finite group and P be a Sylow p-subgroup of G, where p is the smallest prime divisor of |G|. Suppose that P has a non-trivial proper subgroup D such that all subgroups E of P with order |D| and 2|D| (if P is a non-abelian 2-group, |P : D| > 2 and there exists D1 ⊴ E ≤ P with 2|D1| = |D| and E/D1 is cyclic of order 4) have p-nilpotent supplement or weak -supplement in G, then G is p-nilpotent.

ON MINIMAL DEGREES OF PERMUTATION REPRESENTATIONS OF ABELIAN QUOTIENTS OF FINITE GROUPS

2011 ◽  
Vol 84 (3) ◽  
pp. 408-413 ◽  
Author(s):  
CLARA FRANCHI
Keyword(s):  
Finite Group ◽  
Maximal Subgroup ◽  
Finite Groups ◽  
Minimum Degree ◽  
Permutation Representation ◽  
A Finite Group

AbstractFor a finite group G, we denote by μ(G) the minimum degree of a faithful permutation representation of G. We prove that if G is a finite p-group with an abelian maximal subgroup, then μ(G/G′)≤μ(G).

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