On Finite-By-Nilpotent Groups (original) (raw)

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.

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.

A criterion for metanilpotency of a finite group

Journal of Group Theory

We prove that the kth term of the lower central series of a finite group G is nilpotent if and only if {|ab|=|a||b|} for any {\gamma_{k}} -commutators {a,b\in G} of coprime orders.

ON THE FINITENESS PROPERTIES OF GROUPS

For an automorphism ϕ of the group G, the connection between the centralizer C G (ϕ) and the commutator [G, ϕ] is investigated and as a consequence of the Schur theorem it is shown that if G/C G (ϕ) and G ′ are both finite, then so is [G, ϕ].

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]

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.

Bounding the Order of the Nilpotent Residual of a Finite Group

Bulletin of the Australian Mathematical Society

The last term of the lower central series of a finite group$G$is called the nilpotent residual. It is usually denoted by$\unicode[STIX]{x1D6FE}_{\infty }(G)$. The lower Fitting series of$G$is defined by$D_{0}(G)=G$and$D_{i+1}(G)=\unicode[STIX]{x1D6FE}_{\infty }(D_{i}(G))$for$i=0,1,2,\ldots \,$. These subgroups are generated by so-called coprime commutators$\unicode[STIX]{x1D6FE}_{k}^{\ast }$and$\unicode[STIX]{x1D6FF}_{k}^{\ast }$in elements of$G$. More precisely, the set of coprime commutators$\unicode[STIX]{x1D6FE}_{k}^{\ast }$generates$\unicode[STIX]{x1D6FE}_{\infty }(G)$whenever$k\geq 2$while the set$\unicode[STIX]{x1D6FF}_{k}^{\ast }$generates$D_{k}(G)$for$k\geq 0$. The main result of this article is the following theorem: let$m$be a positive integer and$G$a finite group. Let$X\subset G$be either the set of all$\unicode[STIX]{x1D6FE}_{k}^{\ast }$-commutators for some fixed$k\geq 2$or the set of all$\unicode[STIX]{x1D6FF}_{k}^{\ast }$-commutators for some fixed$k\geq 1$. Suppose ...