scholarly journals Formations of Π-soluble groups

1969 ◽  
Vol 10 (1-2) ◽  
pp. 241-250 ◽  
Author(s):  
H. Lausch
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Soluble Group ◽  
Composition Series ◽  
Soluble Groups ◽  
Hall Subgroups ◽  
A Finite Group

The theory of formations of soluble groups, developed by Gaschütz [4], Carter and Hawkes[1], provides fairly general methods for investigating canonical full conjugate sets of subgroups in finite, soluble groups. Those methods, however, cannot be applied to the class of all finite groups, since strong use was made of the Theorem of Galois on primitive soluble groups. Nevertheless, there is a possiblity to extend the results of the above mentioned papers to the case of Π-soluble groups as defined by Čunihin [2]. A finite group G is called Π-soluble, if, for a given set it of primes, the indices of a composition series of G are either primes belonging to It or they are not divisible by any prime of Π In this paper, we shall frequently use the following result of Čunihin [2]: Ift is a non-empty set of primes, Π′ its complement in the set of all primes, and G is a Π-soluble group, then there always exist Hall Π-subgroups and Hall ′-subgroups, constituting single conjugate sets of subgroups of G respectively, each It-subgroup of G contained in a Hall Π-subgroup of G where each ′-subgroup of G is contained in a Hall Π′-subgroup of G. All groups considered in this paper are assumed to be finite and Π-soluble. A Hall Π-subgroup of a group G will be denoted by G.

scholarly journals Automorphisms of Finite Groups and their Fixed-Point Groups

1969 ◽  
Vol 9 (3-4) ◽  
pp. 467-477 ◽  
Author(s):  
J. N. Ward
Keyword(s):  
Fixed Point ◽  
Finite Group ◽  
Abelian Group ◽  
Finite Groups ◽  
Upper Bound ◽  
Prime Order ◽  
Soluble Group ◽  
Point Groups ◽  
Derived Series ◽  
A Finite Group

Let G denote a finite group with a fixed-point-free automorphism of prime order p. Then it is known (see [3] and [8]) that G is nilpotent of class bounded by an integer k(p). From this it follows that the length of the derived series of G is also bounded. Let l(p) denote the least upper bound of the length of the derived series of a group with a fixed-point-free automorphism of order p. The results to be proved here may now be stated: Theorem 1. Let G denote a soluble group of finite order and A an abelian group of automorphisms of G. Suppose that (a) ∣G∣ is relatively prime to ∣A∣; (b) GAis nilpotent and normal inGω, for all ω ∈ A#; (c) the Sylow 2-subgroup of G is abelian; and (d) if q is a prime number andqk+ 1 divides the exponent of A for some integer k then the Sylow q-subgroup of G is abelian.

On residuals of finite groups

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Stefanos Aivazidis ◽  
Thomas Müller
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Nilpotent Groups ◽  
Simple Groups ◽  
Fitting Subgroup ◽  
Soluble Groups ◽  
A Finite Group ◽  
Fitting Formation ◽  
Original Theorem

Abstract Theorem C in [S. Dolfi, M. Herzog, G. Kaplan and A. Lev, The size of the solvable residual in finite groups, Groups Geom. Dyn. 1 (2007), 4, 401–407] asserts that, in a finite group with trivial Fitting subgroup, the size of the soluble residual of the group is bounded from below by a certain power of the group order and that the inequality is sharp. Inspired by this result and some of the arguments in the above article, we establish the following generalisation: if 𝔛 is a subgroup-closed Fitting formation of full characteristic which does not contain all finite groups and X ¯ \overline{\mathfrak{X}} is the extension-closure of 𝔛, then there exists an (explicitly known and optimal) constant 𝛾 depending only on 𝔛 such that, for all non-trivial finite groups 𝐺 with trivial 𝔛-radical, | G X ¯ | > | G | γ \lvert G^{\overline{\mathfrak{X}}}\rvert>\lvert G\rvert^{\gamma} , where G X ¯ G^{\overline{\mathfrak{X}}} is the X ¯ \overline{\mathfrak{X}} -residual of 𝐺. When X = N \mathfrak{X}=\mathfrak{N} , the class of finite nilpotent groups, it follows that X ¯ = S \overline{\mathfrak{X}}=\mathfrak{S} , the class of finite soluble groups; thus we recover the original theorem of Dolfi, Herzog, Kaplan, and Lev. In the last section of our paper, building on J. G. Thompson’s classification of minimal simple groups, we exhibit a family of subgroup-closed Fitting formations 𝔛 of full characteristic such that S ⊂ X ¯ ⊂ E \mathfrak{S}\subset\overline{\mathfrak{X}}\subset\mathfrak{E} , where 𝔈 denotes the class of all finite groups, thus providing applications of our main result beyond the reach of the above theorem.

N-critical graph of finite groups

2021 ◽  
Author(s):  
Viachaslau I. Murashka
Keyword(s):  
Finite Group ◽  
Directed Graph ◽  
Finite Groups ◽  
Soluble Group ◽  
Prime Divisors ◽  
Nilpotent Length ◽  
Critical Graph ◽  
Vertex Set ◽  
A Finite Group

A Schmidt [Formula: see text]-group is a non-nilpotent [Formula: see text]-group whose proper subgroups are nilpotent and which has the normal Sylow [Formula: see text]-subgroup. The [Formula: see text]-critical graph [Formula: see text] of a finite group [Formula: see text] is a directed graph on the vertex set [Formula: see text] of all prime divisors of [Formula: see text] and [Formula: see text] is an edge of [Formula: see text] if and only if [Formula: see text] has a Schmidt [Formula: see text]-subgroup. The bounds of the nilpotent length of a soluble group are obtained in terms of its [Formula: see text]-critical graph. The structure of a soluble group with given [Formula: see text]-critical graph is obtained in terms of commutators. The connections between [Formula: see text]-critical and other graphs (Sylow, soluble, prime, commuting) of finite groups are found.

scholarly journals Products of Finite Connected Subgroups

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1498
Author(s):  
María Pilar Gállego ◽  
Peter Hauck ◽  
Lev S. Kazarin ◽  
Ana Martínez-Pastor ◽  
María Dolores Pérez-Ramos
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Soluble Groups ◽  
Derived Subgroup ◽  
A Finite Group

For a non-empty class of groups L, a finite group G=AB is said to be an L-connected product of the subgroups A and B if ⟨a,b⟩∈L for all a∈A and b∈B. In a previous paper, we prove that, for such a product, when L=S is the class of finite soluble groups, then [A,B] is soluble. This generalizes the theorem of Thompson that states the solubility of finite groups whose two-generated subgroups are soluble. In the present paper, our result is applied to extend to finite groups previous research about finite groups in the soluble universe. In particular, we characterize connected products for relevant classes of groups, among others, the class of metanilpotent groups and the class of groups with nilpotent derived subgroup. Additionally, we give local descriptions of relevant subgroups of finite groups.

scholarly journals On Hall subgroups of a finite group

2013 ◽  
Vol 11 (7) ◽  
Author(s):  
Wenbin Guo ◽  
Alexander Skiba
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Hall Subgroups ◽  
A Finite Group ◽  
New Criteria

AbstractNew criteria of existence and conjugacy of Hall subgroups of finite groups are given.

Finite groups with permutable complete Wielandt sets of subgroups

2015 ◽  
Vol 18 (2) ◽  
Author(s):  
Wenbin Guo ◽  
Alexander N. Skiba
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Hall Subgroups ◽  
A Finite Group

AbstractLet 𝒲 be a set of nilpotent Hall subgroups of a finite group

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.

A Note on the σ-Covers of Finite Soluble Groups

2005 ◽  
Vol 12 (04) ◽  
pp. 691-697 ◽  
Author(s):  
Alireza Jamali ◽  
Hamid Mousavi
Keyword(s):  
Finite Group ◽  
Positive Integer ◽  
Soluble Group ◽  
Maximal Subgroups ◽  
Finite Soluble Group ◽  
Soluble Groups ◽  
A Finite Group

A cover for a finite group is a set of proper subgroups whose union is the whole group. For a finite group G, we denote by σ (G) the least positive integer n such that G has a cover of size n. This paper deals with the covers of a finite soluble group G having σ (G) elements of maximal subgroups.

scholarly journals SOLVABILITY OF FINITE GROUPS VIA CONDITIONS ON PRODUCTS OF 2-ELEMENTS AND ODD p-ELEMENTS

2010 ◽  
Vol 82 (2) ◽  
pp. 265-273 ◽  
Author(s):  
GIL KAPLAN ◽  
DAN LEVY
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
London Math ◽  
Characteristic Property ◽  
Soluble Groups ◽  
P Elements ◽  
Odd Order ◽  
Solvability Criterion ◽  
A Finite Group

AbstractWe observe that a solvability criterion for finite groups, conjectured by Miller [The product of two or more groups, Trans. Amer. Math. Soc.12 (1911)] and Hall [A characteristic property of soluble groups, J. London Math. Soc.12 (1937)] and proved by Thompson [Nonsolvable finite groups all of whose local subgroups are solvable, Bull. Amer. Math. Soc.74(3) (1968)], can be sharpened as follows: a finite group is nonsolvable if and only if it has a nontrivial 2-element and an odd p-element, such that the order of their product is not divisible by either 2 or p. We also prove a solvability criterion involving conjugates of odd p-elements. Finally, we define, via a condition on products of p-elements with p′-elements, a formation Pp,p′, for each prime p. We show that P2,2′ (which contains the odd-order groups) is properly contained in the solvable formation.

scholarly journals Finite soluble groups have large centralisers

1987 ◽  
Vol 35 (2) ◽  
pp. 291-298 ◽  
Author(s):  
John Cossey
Keyword(s):  
Finite Group ◽  
Soluble Group ◽  
Central Element ◽  
Finite Soluble Group ◽  
Soluble Groups ◽  
A Finite Group

We say that a finite group G has a large centraliser if G contains a non-central element x with |CG (x)| > |G|½. We prove that every finite soluble group has a large centraliser, confirming a conjecture of Bertram and Herzog.

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