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.

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.

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.

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 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).

New characterizations of p-soluble and p-supersoluble finite groups

2012 ◽  
Vol 49 (3) ◽  
pp. 390-405
Author(s):  
Wenbin Guo ◽  
Alexander Skiba
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Maximal Subgroup ◽  
Finite Groups ◽  
Prime Order ◽  
Cyclic Subgroup ◽  
Sylow Subgroups ◽  
Quasinormal Subgroup ◽  
A Finite Group

Let G be a finite group and H a subgroup of G. H is said to be S-quasinormal in G if HP = PH for all Sylow subgroups P of G. Let HsG be the subgroup of H generated by all those subgroups of H which are S-quasinormal in G and HsG the intersection of all S-quasinormal subgroups of G containing H. The symbol |G|p denotes the order of a Sylow p-subgroup of G. We prove the followingTheorem A. Let G be a finite group and p a prime dividing |G|. Then G is p-supersoluble if and only if for every cyclic subgroup H ofḠ (G) of prime order or order 4 (if p = 2), Ḡhas a normal subgroup T such thatHsḠandH∩T=HsḠ∩T.Theorem B. A soluble finite group G is p-supersoluble if and only if for every 2-maximal subgroup E of G such that Op′ (G) ≦ E and |G: E| is not a power of p, G has an S-quasinormal subgroup T with cyclic Sylow p-subgroups such that EsG = ET and |E ∩ T|p = |EsG ∩ T|p.Theorem C. A finite group G is p-soluble if for every 2-maximal subgroup E of G such that Op′ (G) ≦ E and |G: E| is not a power of p, G has an S-quasinormal subgroup T such that EsG = ET and |E ∩ Tp = |EsG ∩ T|p.

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.

Finite groups with short nonnormal chains

1974 ◽  
Vol 18 (1) ◽  
pp. 111-118
Author(s):  
Armond E. Spencer
Keyword(s):  
Finite Group ◽  
Maximal Subgroup ◽  
Finite Groups ◽  
A Finite Group ◽  
Subnormal Subgroups

This note is a continuation of the author's work [6], describing the structure of a finite group given some information about the distribution of the subnormal subgroups in the lattice of all subgoups.DEFINITION. An upper chain of length n in the finite group G is a sequence of subgroups of G; G = Go > G1 > … > Gn, such that for each i, Gi is a maximal subgroup of Gi-1. Let h(G) = n if every upper chain in G of length n contains a proper ( ≠ G) subnormal entry, and there is at least one upper chain in G of length (n – 1) which contains no proper subnormal entry.

scholarly journals Nilpotent injectors in finite groups

1985 ◽  
Vol 32 (2) ◽  
pp. 293-297 ◽  
Author(s):  
Peter Förster
Keyword(s):  
Finite Group ◽  
Conjugacy Class ◽  
Maximal Subgroup ◽  
Finite Groups ◽  
Nilpotent Groups ◽  
Fitting Class ◽  
Basic Properties ◽  
A Finite Group ◽  
Class F

Nilpotent injectors exist in all finite groups.For every Fitting class F of finite groups (see [2]), InjF(G) denotes the set of all H ≤ G such that for each N ⊴ ⊴ G , H ∩ N is an F -maximal subgroup of N (that is, belongs to F and i s maximal among the subgroups of N with this property). Let W and N* denote the Fitting class of all nilpotent and quasi-nilpotent groups, respectively. (For the basic properties of quasi-nilpotent groups, and of the N*-radical F*(G) of a finite group G3 the reader is referred to [5].,X. %13; we shall use these properties without further reference.) Blessenohl and H. Laue have shown in CJ] that for every finite group G, InjN*(G) = {H ≤ G | H ≥ F*(G) N*-maximal in G} is a non-empty conjugacy class of subgroups of G. More recently, Iranzo and Perez-Monasor have verified InjN(G) ≠ Φ for all finite groups G satisfying G = CG(E(G))E(G) (see [6]), and have extended this result to a somewhat larger class M of finite groups C(see [7]). One checks, however, that M does not contain all finite groups; for example, S5 ε M.

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