The nilpotent length of finite soluble groups (original) (raw)
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Bulletin of the Australian Mathematical Society, 1969
In a finite soluble group G, the Fitting (or nilpotency) length h(G) can be considered as a measure for how strongly G deviates from being nilpotent. As another measure for this, the number v(G) of conjugacy classes of the maximal nilpotent subgroups of G may be taken. It is shown that there exists an integer-valued function f on the set of positive integers such that h(G) ≦ f(v(G)) for all finite (soluble) groups of odd order. Moreover, if all prime divisors of the order of G are greater than v(G)(v(G) - l)/2, then h(G) ≦3. The bound f(v(G)) is just of qualitative nature and by far not best possible. For v(G) = 2, h(G) = 3, some statements are made about the structure of G.
On the p-length of some finite p-soluble groups
Israel Journal of Mathematics, 2014
The main aim of this paper is to give structural information of a finite group of minimal order belonging to a subgroup-closed class of finite groups and whose p-length is greater than 1, p a prime number. Alternative proofs and improvements of recent results about the influence of minimal p-subgroups on the p-nilpotence and p-length of a finite group arise as consequences of our study.
Glasgow Mathematical Journal
Abstarct Let γ n = [x1,…,x n ] be the nth lower central word. Denote by X n the set of γ n -values in a group G and suppose that there is a number m such that ∣gXn∣lem|{g^{{X_n}}}| \le m∣gXn∣lem for each g ∈ G. We prove that γn+1(G) has finite (m, n) -bounded order. This generalizes the much-celebrated theorem of B. H. Neumann that says that the commutator subgroup of a BFC-group is finite.
On the σ-Length of Maximal Subgroups of Finite σ-Soluble Groups
Mathematics
Let σ={σi:i∈I} be a partition of the set P of all prime numbers and let G be a finite group. We say that G is σ-primary if all the prime factors of |G| belong to the same member of σ. G is said to be σ-soluble if every chief factor of G is σ-primary, and G is σ-nilpotent if it is a direct product of σ-primary groups. It is known that G has a largest normal σ-nilpotent subgroup which is denoted by Fσ(G). Let n be a non-negative integer. The n-term of the σ-Fitting series of G is defined inductively by F0(G)=1, and Fn+1(G)/Fn(G)=Fσ(G/Fn(G)). If G is σ-soluble, there exists a smallest n such that Fn(G)=G. This number n is called the σ-nilpotent length of G and it is denoted by lσ(G). If F is a subgroup-closed saturated formation, we define the σ-F-lengthnσ(G,F) of G as the σ-nilpotent length of the F-residual GF of G. The main result of the paper shows that if A is a maximal subgroup of G and G is a σ-soluble, then nσ(A,F)=nσ(G,F)−i for some i∈{0,1,2}.
On finite products of nilpotent groups
Archiv der Mathematik, 1994
i. Introduetion. A well-known theorem of Kegel [7] and Wielandt [9] states the solubility of every finite group G = AB which is the product of two nilpotent subgroups A and B; see [1], Theorem 2.4.3. In order to determine the structure of these groups it is of interest to know which subgroups of G are conjugate (or at least isomorphic) to a subgroup that inherits the factorization. A subgroup S of the factorized group G = AB is called prefactorized if S = (A c~ S) (B ~ S), it is called factorized if, in addition, S contains the intersection A c~ B. Generally, even characteristic subgroups of G are not prefactorized, as can be seen e.g. from Examples 1 and 2 below.
On the Nilpotent Length of Polycyclic Groups
Journal of Algebra, 1998
Let G be a polycyclic group. We prove that if the nilpotent length of each finite quotient of G is bounded by a fixed integer n, then the nilpotent length of G is at most n. The case n s 1 is a well-known result of Hirsch. As a consequence, we obtain that if the nilpotent length of each 2-generator subgroup is at most n, then the nilpotent length of G is at most n. A more precise result in the case n s 2 permits us to prove that if each 3-generator subgroup is abelian-by-nilpotent, then G is abelian-by-nilpotent. Furthermore, we show that the nilpotent length of G equals the nilpotent length of the quotient of G by its Frattini subgroup. ᮊ 1998 Academic Press *
On Torsion-by-Nilpotent Groups
Journal of Algebra, 2001
Let C C be a class of groups, closed under taking subgroups and quotients. We prove that if all metabelian groups of C C are torsion-by-nilpotent, then all soluble groups of C C are torsion-by-nilpotent. From that, we deduce the following conse-Ž quence, similar to a well-known result of P. Hall 1958, Illinois J. Math. 2,. 787᎐801 : if H is a normal subgroup of a group G such that H and GrHЈ are Ž. Ž. locally finite-by-nilpotent, then G is locally finite-by-nilpotent. We give an Ž. example showing that this last statement is false when '' locally finite-by-nilpotent'' is replaced with ''torsion-by-nilpotent.''
Capability of finite nilpotent groups of class 2 with cyclic Frattini subgroups
Journal of Advanced Research in Pure Mathematics, 2014
A group is called capable if it is a central factor group. Let N denote the set of all finite groups of nilpotency class 2 whose derived subgroups be cyclic and coincide with their Frattini subgroups. This paper is organized to provide the explicit structures of capable groups in N.
On nilpotency of higher commutator subgroups of a finite soluble group
Archiv der Mathematik, 2020
Let G be a finite soluble group and G (k) the kth term of the derived series of G. We prove that G (k) is nilpotent if and only if |ab| = |a||b| for any δ k-values a, b ∈ G of coprime orders. In the course of the proof we establish the following result of independent interest: Let P be a Sylow p-subgroup of G. Then P ∩ G (k) is generated by δ k-values contained in P (Lemma 2.5). This is related to the so-called Focal Subgroup Theorem. Let G be a finite group in which |ab| = |a||b| whenever the elements a, b have coprime orders. Then G is nilpotent. Here the symbol |x| stands for the order of an element x in a group G. In [2] a similar sufficient condition for nilpotency of the commutator subgroup G ′ was established. Let G be a finite group in which |ab| = |a||b| whenever the elements a, b are commutators of coprime orders. Then G ′ is nilpotent. Of course, the conditions in both above results are also necessary for the nilpotency of G and G ′ , respectively. More recently, in [3] the above results were extended as follows. Given an integer k ≥ 1, the word γ k = γ k (x 1 ,. .. , x k) is defined inductively by the formulae γ 1 = x 1 , and γ k = [γ k−1 , x k ] = [x 1 ,. .. , x k ] for k ≥ 2.
On Groups with All Subgroups Subnormal or Soluble of Bounded Derived Length
arXiv: Group Theory, 2012
In this paper, we deal with locally graded groups whose subgroups are either subnormal or soluble of bounded derived length, say d. In particular, we prove that every locally (soluble-by-finite) group with this property is either soluble or an extension of a soluble group of derived length at most d by a finite group, which fits between a minimal simple group and its automorphism group. We also classify all the finite non-abelian simple groups whose proper subgroups are metabelian.