Minimal non-𝒬𝒮-groups

2019 ◽  
Vol 29 (04) ◽  
pp. 713-722
Author(s):  
Huaguo Shi ◽  
Zhangjia Han ◽  
Heng Lv ◽  
Longhui Zhang
Keyword(s):  
Finite Group ◽  
Proper Subgroup ◽  
A Finite Group

A finite group [Formula: see text] is called a [Formula: see text]-group if every proper subgroup of [Formula: see text] is either quasi-normal or self-normal in [Formula: see text]. In this paper, the authors classify the non-[Formula: see text]-groups whose proper subgroups are all [Formula: see text]-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.

The Influence of Generalized Frattini Subgroups on the Solvability of a Finite Group

1970 ◽  
Vol 22 (1) ◽  
pp. 41-46 ◽  
Author(s):  
James C. Beidleman
Keyword(s):  
Finite Group ◽  
Solvable Group ◽  
Proper Subgroup ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Fitting Subgroup ◽  
Frattini Subgroup ◽  
A Finite Group ◽  
Necessary And Sufficient ◽  
Equivalent Conditions

1. The Frattini and Fitting subgroups of a finite group G have been useful subgroups in establishing necessary and sufficient conditions for G to be solvable. In [1, pp. 657-658, Theorem 1], Baer used these subgroups to establish several very interesting equivalent conditions for G to be solvable. One of Baer's conditions is that ϕ(S), the Frattini subgroup of S, is a proper subgroup of F(S), the Fitting subgroup of S, for each subgroup S ≠ 1 of G. Using the Fitting subgroup and generalized Frattini subgroups of certain subgroups of G we provide certain equivalent conditions for G to be a solvable group. One such condition is that F(S) is not a generalized Frattini subgroup of S for each subgroup S ≠ 1 of G. Our results are given in Theorem 1.

Generalized Frattini Subgroups of Finite Groups. II

1969 ◽  
Vol 21 ◽  
pp. 418-429 ◽  
Author(s):  
James C. Beidleman
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Proper Subgroup ◽  
Subnormal Subgroup ◽  
Normal Subgroups ◽  
Frattini Subgroup ◽  
A Finite Group ◽  
Proper Normal Subgroup ◽  
Equivalent Conditions

The theory of generalized Frattini subgroups of a finite group is continued in this paper. Several equivalent conditions are given for a proper normal subgroup H of a finite group G to be a generalized Frattini subgroup of G. One such condition on H is that K is nilpotent for each normal subgroup K of G such that K/H is nilpotent. From this result, it follows that the weakly hyper-central normal subgroups of a finite non-nilpotent group G are generalized Frattini subgroups of G.Let H be a generalized Frattini subgroup of G and let K be a subnormal subgroup of G which properly contains H. Then H is a generalized Frattini subgroup of K.Let ϕ(G) be the Frattini subgroup of G. Suppose that G/ϕ(G) is nonnilpotent, but every proper subgroup of G/ϕ(G) is nilpotent. Then ϕ(G) is the unique maximal generalized Frattini subgroup of G.

On the ℱϕ-Hypercentre of Finite Groups

2014 ◽  
Vol 57 (3) ◽  
pp. 648-657 ◽  
Author(s):  
Juping Tang ◽  
Long Miao
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Maximal Subgroup ◽  
Finite Groups ◽  
Sylow Subgroup ◽  
Proper Subgroup ◽  
Normal Subgroups ◽  
Chief Factors ◽  
A Finite Group

AbstractLet G be a finite group and let ℱ be a class of groups. Then Zℱϕ(G) is the ℱϕ-hypercentre of G, which is the product of all normal subgroups of G whose non-Frattini G-chief factors are ℱ-central in G. A subgroup H is called ℳ-supplemented in a finite group G if there exists a subgroup B of G such that G = HB and H1B is a proper subgroup of G for any maximal subgroup H1 of H. The main purpose of this paper is to prove the following: Let E be a normal subgroup of a group G. Suppose that every noncyclic Sylow subgroup P of F*(E) has a subgroup D such that 1 < |D| < |P| and every subgroup H of P with order |H| = |D| is 𝓜-supplemented in G, then E ≤ Zuϕ(G).

On the weakly monomial subgroups of finite simple groups

2019 ◽  
Vol 18 (02) ◽  
pp. 1950037
Author(s):  
Shuqin Dong ◽  
Hongfei Pan ◽  
Feng Tang
Keyword(s):  
Finite Group ◽  
Proper Subgroup ◽  
Maximal Subgroups ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Group A ◽  
A Finite Group

Let [Formula: see text] be a finite group. A proper subgroup [Formula: see text] of [Formula: see text] is said to be weakly monomial if the order of [Formula: see text] satisfies [Formula: see text]. In this paper, we determine all the weakly monomial maximal subgroups of finite simple groups.

ON A THEOREM OF HUPPERT

2011 ◽  
Vol 10 (02) ◽  
pp. 295-301
Author(s):  
JIANGTAO SHI ◽  
CUI ZHANG
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Soluble Group ◽  
Proper Subgroup ◽  
Maximal Subgroups ◽  
Frattini Subgroup ◽  
A Finite Group ◽  
Group 1

A well-known theorem of Huppert states that a finite group is soluble if its every proper subgroup is supersoluble. In this paper, we proved the following result: let G be a finite group. (1) If G has exactly n non-supersoluble proper subgroups, where 0 ≤ n ≤ 7 and n ≠ 5, then G is soluble. (2) G is a non-soluble group with exactly five non-supersoluble proper subgroups if and only if all non-supersoluble proper subgroups are conjugate maximal subgroups and G/Φ(G) ≅ A5, where Φ(G) is the Frattini subgroup of G. Furthermore, we also considered the influence of the number of non-abelian proper subgroups on the solubility of finite groups.

scholarly journals ON THE PRIME GRAPH OF SIMPLE GROUPS

2014 ◽  
Vol 91 (2) ◽  
pp. 227-240 ◽  
Author(s):  
TIMOTHY C. BURNESS ◽  
ELISA COVATO
Keyword(s):  
Finite Group ◽  
Proper Subgroup ◽  
Prime Graph ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Prime Divisors ◽  
Prime Graphs ◽  
Vertex Set ◽  
A Finite Group

AbstractLet $G$ be a finite group, let ${\it\pi}(G)$ be the set of prime divisors of $|G|$ and let ${\rm\Gamma}(G)$ be the prime graph of $G$. This graph has vertex set ${\it\pi}(G)$, and two vertices $r$ and $s$ are adjacent if and only if $G$ contains an element of order $rs$. Many properties of these graphs have been studied in recent years, with a particular focus on the prime graphs of finite simple groups. In this note, we determine the pairs $(G,H)$, where $G$ is simple and $H$ is a proper subgroup of $G$ such that ${\rm\Gamma}(G)={\rm\Gamma}(H)$.

scholarly journals A nonabelian Frobenius–Wielandt complement

1988 ◽  
Vol 31 (1) ◽  
pp. 67-69 ◽  
Author(s):  
Alberto Espuelas
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Proper Subgroup ◽  
A Finite Group ◽  
Proper Normal Subgroup

We recall the following definition (see [1]):A finite group G is said to be a Frobenius–Wielandt group provided that there exists a proper subgroup H of G and a proper normal subgroup N of H such that H∩Hg≦N if g∈G–H. Then H/N is said to be the complement of (G, H, N) (see [1] for more details and notation).

scholarly journals A classification of groups with a centralizer condition II

1977 ◽  
Vol 16 (1) ◽  
pp. 55-60 ◽  
Author(s):  
Zvi Arad ◽  
Marcel Herzog
Keyword(s):  
Finite Group ◽  
Proper Subgroup ◽  
Group A ◽  
A Finite Group

Let G be a finite group. A nontrivial proper subgroup M of G is called a CC-subgpoup if M contains the centralizer in G of each of its nonidentity elements. In this paper groups containing a CC-subgroup of order divisible by 3 are completely determined.

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