A criterion for metanilpotency of a finite group (original) (raw)
Related papers
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 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.
Coprime commutators in finite groups
2018
Let G be a finite group and let k ≥ 2. We prove that the coprime subgroup γ_k^*(G) is nilpotent if and only if |xy|=|x||y| for any γ_k^*-commutators x,y ∈ G of coprime orders (Theorem A). Moreover, we show that the coprime subgroup δ_k^*(G) is nilpotent if and only if |ab|=|a||b| for any powers of δ_k^*-commutators a,b∈ G of coprime orders (Theorem B).
On the Nilpotency of a Pair of Groups
Southeast Asian Bulletin of …, 2012
This paper is devoted to suggest that the extensive theory of nilpotency, upper and lower central series of groups could be extended in an interesting and useful way to a theory for pairs of groups. Also this yields some information on nilpotent groups.
Nilpotency: A Characterization Of The Finite p-Groups
Journal of Mathematics , 2017
Abstract As parts of the characterizations of the finite p-groups is the fact that every finite p-group is NILPOTENT. Hence, there exists a derived series (Lower Central) which terminates at e after a finite number of steps. Suppose that G is a p-group of class at least m ≥ 3. Then L m-1G is abelian and hence G possesses a characteristic abelian subgroup which is not contained in Z(G). If L 3(G) = 1 such that pm is the highest order of an element of G/L2 (G) (where G is any p-group) then no element of L2(G) has an order higher than pm. [1]
Two characterizations of finite nilpotent groups
Journal of Group Theory, 2018
In this note we give two characterizations of finite nilpotent groups. First, we show that a finite group G is not p-nilpotent if and only if it contains two elements of order q k {q^{k}} , for q a prime different than p, whose product has order p or possibly 4 if p = 2 {p=2} . We also show that the set of words on two variables where the total degree of each variable is ± 1 {\pm 1} can be used to characterize finite nilpotent groups. Using this characterization we show that if a finite group is not nilpotent, then there is a word map of specified form for which the corresponding probability distribution is not uniform.
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.
Introduction to Nilpotent Groups
The Theory of Nilpotent Groups, 2017
is an ascending series (or an ascending chain of subgroups). (ii) If G i G j for 1 Ä i Ä j; then G 1 G 2 G 3 (2.2) is a descending series (or a descending chain of subgroups). An ascending series may not reach G: If it does, then we say that the series terminates in G. Similarly, a descending series which reaches the identity is said to terminate in the identity. If there exists an integer m > 1 such that G m 1 ¤ G m and G m D G mC1 D G mC2 D in either (2.1) or (2.2), then the series is said to stabilize in G m. 2.1.2 Definition of a Nilpotent Group Definition 2.3 A group G is called nilpotent if it has a normal series