Notes on Splitting Extensions of Groups

1968 ◽  
Vol 11 (3) ◽  
pp. 371-374 ◽  
Author(s):  
C.Y. Tang
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Commutator Subgroup ◽  
Similar Type ◽  
Frattini Subgroup ◽  
Abelian Normal Subgroup ◽  
A Finite Group

In [1] Gaschütz has shown that a finite group G splits over an abelian normal subgroup N if its Frattini subgroup ϕ(G) intersects N trivially. When N is a non-abelian nilpotent normal subgroup of G the condition ϕ(G)∩ N = 1 cannot be satisfied: for if N is non-abelian then the commutator subgroup C(N) of N is non-trivial. Now N is nilpotent, whence 1 ≠ C(N)⊂ϕ(N). Since G is a finite group, therefore, by (3, theorem 7.3.17) ϕ⊂ϕ(G). It follows that ϕ(G) ∩ N ≠ 1. Thus the condition ϕ(G) ∩ N = 1 must be modified. In §1 we shall derive some similar type of conditions for G to split over N when the restriction of N being an abelian normal subgroup is removed. In § 2 we shall give a characterization of splitting extensions of N in which every subgroup splits over its intersection with N.

scholarly journals A generalized Frattini subgroup of a finite group

1989 ◽  
Vol 12 (2) ◽  
pp. 263-266
Author(s):  
Prabir Bhattacharya ◽  
N. P. Mukherjee
Keyword(s):  
Finite Group ◽  
Maximal Subgroups ◽  
Frattini Subgroup ◽  
Definition Of ◽  
A Finite Group

For a finite group G and an arbitrary prime p, letSP(G)denote the intersection of all maximal subgroups M of G such that [G:M] is both composite and not divisible by p; if no such M exists we setSP(G)= G. Some properties of G are considered involvingSP(G). In particular, we obtain a characterization of G when each M in the definition ofSP(G)is nilpotent.

A result on p-hypercyclically embedded subgroups

2019 ◽  
Vol 18 (03) ◽  
pp. 1950043
Author(s):  
Changwen Li ◽  
Jianhong Huang ◽  
Bin Hu
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
A Finite Group

In this paper, a new characterization of [Formula: see text]-hypercyclical embeddability of a normal subgroup of a finite group is obtained based on the notion of [Formula: see text]-subgroups and some known results are generalized and extended.

Critical maximal subgroups and conjugacy of supplements in finite soluble groups

2018 ◽  
Vol 21 (1) ◽  
pp. 45-63
Author(s):  
Barbara Baumeister ◽  
Gil Kaplan
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Conjugacy Class ◽  
Maximal Subgroups ◽  
Abelian Normal Subgroup ◽  
Soluble Groups ◽  
A Finite Group

AbstractLetGbe a finite group with an abelian normal subgroupN. When doesNhave a unique conjugacy class of complements inG? We consider this question with a focus on properties of maximal subgroups. As corollaries we obtain Theorems 1.6 and 1.7 which are closely related to a result by Parker and Rowley on supplements of a nilpotent normal subgroup [3, Theorem 1]. Furthermore, we consider families of maximal subgroups ofGclosed under conjugation whose intersection equals{\Phi(G)}. In particular, we characterize the soluble groups having a unique minimal family with this property (Theorem 2.3, Remark 2.4). In the case when{\Phi(G)=1}, these are exactly the soluble groups in which each abelian normal subgroup has a unique conjugacy class of complements.

scholarly journals Some New Applications of Weakly H-Embedded Subgroups of Finite Groups

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 158
Author(s):  
Li Zhang ◽  
Li-Jun Huo ◽  
Jia-Bao Liu
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Sylow Subgroup ◽  
Normal Closure ◽  
A Finite Group ◽  
New Applications ◽  
New Criterion

A subgroup H of a finite group G is said to be weakly H -embedded in G if there exists a normal subgroup T of G such that H G = H T and H ∩ T ∈ H ( G ) , where H G is the normal closure of H in G, and H ( G ) is the set of all H -subgroups of G. In the recent research, Asaad, Ramadan and Heliel gave new characterization of p-nilpotent: Let p be the smallest prime dividing | G | , and P a non-cyclic Sylow p-subgroup of G. Then G is p-nilpotent if and only if there exists a p-power d with 1 < d < | P | such that all subgroups of P of order d and p d are weakly H -embedded in G. As new applications of weakly H -embedded subgroups, in this paper, (1) we generalize this result for general prime p and get a new criterion for p-supersolubility; (2) adding the condition “ N G ( P ) is p-nilpotent”, here N G ( P ) = { g ∈ G | P g = P } is the normalizer of P in G, we obtain p-nilpotence for general prime p. Moreover, our tool is the weakly H -embedded subgroup. However, instead of the normality of H G = H T , we just need H T is S-quasinormal in G, which means that H T permutes with every Sylow subgroup of G.

scholarly journals On Finite Quasi-Core-p p-Groups

Symmetry ◽  
2019 ◽  
Vol 11 (9) ◽  
pp. 1147
Author(s):  
Jiao Wang ◽  
Xiuyun Guo
Keyword(s):  
Finite Group ◽  
Positive Integer ◽  
Commutator Subgroup ◽  
Nilpotency Class ◽  
Frattini Subgroup ◽  
Normal Core ◽  
A Finite Group

Given a positive integer n, a finite group G is called quasi-core-n if ⟨ x ⟩ / ⟨ x ⟩ G has order at most n for any element x in G, where ⟨ x ⟩ G is the normal core of ⟨ x ⟩ in G. In this paper, we investigate the structure of finite quasi-core-p p-groups. We prove that if the nilpotency class of a quasi-core-p p-group is p + m , then the exponent of its commutator subgroup cannot exceed p m + 1 , where p is an odd prime and m is non-negative. If p = 3 , we prove that every quasi-core-3 3-group has nilpotency class at most 5 and its commutator subgroup is of exponent at most 9. We also show that the Frattini subgroup of a quasi-core-2 2-group is abelian.

First-order characterization of the radical of a finite group

2009 ◽  
Vol 74 (4) ◽  
pp. 1429-1435 ◽  
Author(s):  
John S. Wilson
Keyword(s):  
Finite Group ◽  
Group Theory ◽  
Normal Subgroup ◽  
First Order ◽  
Order Language ◽  
A Finite Group

AbstractIt is shown that there is a formula σ(g) in the first-order language of group theory with the following property: for every finite group G, the largest soluble normal subgroup of G consists precisely of the elements g of G such that σ(g) holds.

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.

scholarly journals Noether's Problem for p-Groups with an Abelian Subgroup of Index p

2015 ◽  
Vol 22 (spec01) ◽  
pp. 835-848
Author(s):  
Ivo M. Michailov
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Function Field ◽  
Abelian Subgroup ◽  
Normal Subgroups ◽  
Abelian Normal Subgroup ◽  
Root Of Unity ◽  
Type C ◽  
A Finite Group ◽  
Noether’S Problem

Let K be a field and G be a finite group. Let G act on the rational function field K(x(g) : g ∈ G) by K-automorphisms defined by g · x(h) = x(gh) for any g, h ∈ G. Denote by K(G) the fixed field K(x(g) : g ∈ G)G. Noether's problem then asks whether K(G) is rational over K. Let p be an odd prime and let G be a p-group of exponent pe. Assume also that (i) char K = p > 0, or (ii) char K ≠ p and K contains a primitive pe-th root of unity. In this paper we prove that K(G) is rational over K for the following two types of groups: (1) G is a finite p-group with an abelian normal subgroup H of index p such that H is a direct product of normal subgroups of G of the type Cpb × (Cp)c for some b, c with 1 ≤ b and 0 ≤ c; (2) G is any group of order p5 from the isoclinic families with numbers 1, 2, 3, 4, 8 and 9.

ON COMMUTATOR LENGTH OF CERTAIN CLASSES OF SOLVABLE GROUPS

2005 ◽  
Vol 15 (01) ◽  
pp. 143-147 ◽  
Author(s):  
MEHRI AKHAVAN-MALAYERI
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Normal Subgroup ◽  
Solvable Group ◽  
Commutator Subgroup ◽  
Solvable Groups ◽  
Abelian Normal Subgroup ◽  
Number Of Generators ◽  
Solvability Length ◽  
Commutator Length

Let G be a group and G′ be its commutator subgroup. Denote by c(G) the minimal number such that every element of G′ can be expressed as a product of at most c(G) commutators. We find suitable bounds for c(G) when G is a free nilpotent by abelian group. Then we prove that c(G) is finite if G is a n-generator solvable group. And G has a nilpotent by abelian normal subgroup K of finite index. Moreover we have c(G) ≤ s(s + 1)/2 + 72n2 + 47n, where s is the number of generators of K. We also prove that in a solvable group of finite Pruffer rank s every element of its commutator subgroup is equal to a product of at most s(s + 1)/2 + 72s2 + 47s. And finally as a corollary of the above results we show that if A is a normal subgroup of a solvable group G such that G/A is a d-generator finite group. And A has finite Pruffer rank s. Then c(G) ≤ s(s + 1)/2 + 72(s2 + n2) + 47(s + n). The bounds we find are independent of the solvability length of the groups.

scholarly journals On the commuting probability for subgroups of a finite group

2021 ◽  
pp. 1-14
Author(s):  
Eloisa Detomi ◽  
Pavel Shumyatsky
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Commutator Subgroup ◽  
Lower Central Series ◽  
Central Series ◽  
Fitting Subgroup ◽  
Typical Application ◽  
The Generalized Fitting Subgroup ◽  
A Finite Group ◽  
Commuting Probability

Let $K$ be a subgroup of a finite group $G$ . The probability that an element of $G$ commutes with an element of $K$ is denoted by $Pr(K,G)$ . Assume that $Pr(K,G)\geq \epsilon$ for some fixed $\epsilon >0$ . We show that there is a normal subgroup $T\leq G$ and a subgroup $B\leq K$ such that the indices $[G:T]$ and $[K:B]$ and the order of the commutator subgroup $[T,B]$ are $\epsilon$ -bounded. This extends the well-known theorem, due to P. M. Neumann, that covers the case where $K=G$ . We deduce a number of corollaries of this result. A typical application is that if $K$ is the generalized Fitting subgroup $F^{*}(G)$ then $G$ has a class-2-nilpotent normal subgroup $R$ such that both the index $[G:R]$ and the order of the commutator subgroup $[R,R]$ are $\epsilon$ -bounded. In the same spirit we consider the cases where $K$ is a term of the lower central series of $G$ , or a Sylow subgroup, etc.

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