Finite groups in which every nonsolvable maximal subgroup is a Hall subgroup

2014 ◽  
Vol 285 (S1) ◽  
pp. 191-202 ◽  
Author(s):  
V. A. Vedernikov
Keyword(s):  
Maximal Subgroup ◽  
Finite Groups ◽  
Hall Subgroup

scholarly journals Maximal subgroups of finite groups

1990 ◽  
Vol 13 (2) ◽  
pp. 311-314
Author(s):  
S. Srinivasan
Keyword(s):  
Maximal Subgroup ◽  
Finite Groups ◽  
Maximal Subgroups ◽  
Fitting Subgroup ◽  
Small Index

In finite groups maximal subgroups play a very important role. Results in the literature show that if the maximal subgroup has a very small index in the whole group then it influences the structure of the group itself. In this paper we study the case when the index of the maximal subgroups of the groups have a special type of relation with the Fitting subgroup of the group.

Irreducible Automorphisms of Certainp-Groups

1977 ◽  
Vol 29 (2) ◽  
pp. 333-348 ◽  
Author(s):  
D. Ž. Djoković ◽  
J. Malzan
Keyword(s):  
Finite Field ◽  
Maximal Subgroup ◽  
Finite Groups ◽  
General Problem ◽  
The Other ◽  
Frattini Subgroup ◽  
Other Hand

The chief purpose of this paper is to find all pairs (G, θ) whereGis a finite specialp-group, andθis an automorphism ofGacting trivially on the Frattini subgroup and irreducibly on the Frattini quotient. This problem arises in the context of describing finite groups having an abelian maximal subgroup. In fact, we solve a more general problem for a wider class ofp-groups, which we callspecial F-groups,whereFis a finite field of characteristicp.We point out that ifpis odd, then anF-group has exponentp.On the other hand, every special 2-group is also a specialGF(2)-group.

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.

Notes on Solvability and p-Supersolvability of Finite Groups

2019 ◽  
Vol 26 (01) ◽  
pp. 139-146
Author(s):  
Jia Zhang ◽  
Tingting Qiu ◽  
Long Miao ◽  
Juping Tang
Keyword(s):  
Maximal Subgroup ◽  
Finite Groups

A subgroup H of G is called ℳ-supplemented in G if there exists a subgroup B of G such that G = HB and Hi B < G for every maximal subgroup Hi of H. In this paper, we use ℳ-supplemented subgroups to study the structure of finite groups and obtain some new characterization about solvability and p-supersolvability for a fixed prime p. Some results in the literature are corollaries of our theorems.

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).

Finite groups in which every 3-Maximal subgroup commutes with all maximal subgroups

2009 ◽  
Vol 86 (3-4) ◽  
pp. 325-332 ◽  
Author(s):  
Wen-Bin Guo ◽  
E. V. Legchekova ◽  
A. N. Skiba
Keyword(s):  
Maximal Subgroup ◽  
Finite Groups ◽  
Maximal Subgroups

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 Uniquen-Maximal Subgroup

2010 ◽  
Vol 38 (12) ◽  
pp. 4514-4519 ◽  
Author(s):  
Guohua Qian
Keyword(s):  
Maximal Subgroup ◽  
Finite Groups
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