On a Property of Nilpotent Groups

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.

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.

Journal of the Australian Mathematical Society, 1988

Let G be a group factorized by finitely many pairwise permutable nilpotent subgroups. The aim of this paper is to find conditions under which at least one of the factors is contained in a proper normal subgroup of G.

Advances in Mathematics, 1982

If G is a group and x, y ~ G, then [x,y] = x l y-l x y is the commutator of x and y. Set F G = {[x,y][x, y C G} and 2 (G) = n , where n is the smallest integer such that every element of G' is a product of n commutators. The problem of determining when G' = FG (i.e., L(G)= 1) is of particular interest. Fire [2] constructs a group G of order 256 with G' elementary abelian of order 16 and [/'l = 15. Here it is shown that the smallest groups G with G' 4: FG are of order 96. If G = S L , (F) , then Shoda [13] (for F algebraically closed) and Thompson [15] (with some exceptions if F has characteristic 0) show G ' = FG. Similar results are known for A n, the Mathieu groups, the Suzuki groups, and semisimple algebraic groups over algebraically closed fields (cf. [9, 11, 16, 17]). It is still an open question whether G' = F G for all finite simple groups. Isaacs [8] gives examples of finite perfect groups with G' 4:FG. For examples of finite groups with 2(G) arbitrarily large see [5, 10]. In [6], all pairs (m, n) are determined such that there exists a group G with G' ~-C(n) (the cyclic group of order n) and 2(G) > m. In particular if G' is cyclic and either IGI < 240 or [G')< 60, then G' =FG. Dark and Newell [1] investigate similar questions for the other terms in the descending central series. Let Sylo(G) denote the set of Sylow p-subgroups of G and d(G) the minimal number of elements needed to generate G. In this paper, commutator subgroups with rank 2 Sylow p-subgroups are considered, and the following is proved.

Groups, Geometry, and Dynamics, 2014

A recipe for obtaining finitely presented abelian-by-nilpotent groups is given. It relies on a geometric procedure that generalizes the construction of finitely presented metabelian groups introduced by R. Mathematics Subject Classification (2010) 20F16, 20F65. in a series of papers in the late 1990s: in [5], Brookes, Roseblade and Wilson showed that a finitely presented abelian-by-polycyclic group G is necessarily nilpotent-by-nilpotent-byfinite; in then, this conclusion is sharpened to G is nilpotent, by nilpotent of class at most two, by finite.

Algebraic structures and their applications

In this paper, groups with trivial intersection between Frattini and derived subgroups are considered. First, some structural properties of these groups are given in an important special case. Then, some family invariants of each n-isoclinism family of such groups are stated. In particular, an explicit bound for the order of each center factor group in terms of the order of its derived subgroup is also provided.

Journal of Algebra, 1986

Bulletin of The Australian Mathematical Society, 1992

Journal of Pure and Applied Algebra, 1982

Journal of Algebra, 2007

It is proved that a finite group G = AB which is a product of a nilpotent subgroup A and a subgroup B with non-trivial center contains a non-trivial abelian normal subgroup.

Journal of Algebra, 2002

We explore the class of generalized nilpotent groups in the universe c of all radical locally finite groups satisfying min-p for every prime p. We obtain that this class is the natural generalization of the class of finite nilpotent groups from the finite universe to the universe c. Moreover, the structure of-groups is determined explicitly. It is also shown that is a subgroup-closed c-formation and that in every c-group the Fitting subgroup is the unique maximal normal-subgroup.

Mediterranean Journal of Mathematics, 2013

A group G is called 3-abelian if the map x → x 3 is an endomorphism of G and it is called generalized 3-abelian, if there exist elements c1, c2, c3 ∈ G such that the map ϕ : x −→ x c 1 x c 2 x c 3 is an endomorphism of G. Abdollahi, Daoud and Endimioni have proved that a generalized 3-abelian group G is nilpotent of class at most 10. Here, we improve the bound to 3 and we show that the exponent of its derived subgoup is finite and divides 9. We also prove that G is 3-Levi, 9-central, 9-abelian and 3-nilpotent of class at most 2.

2017

We show that relative Property (T) for the abelianization of a nilpotent normal subgroup implies relative Property (T) for the subgroup itself. This and other results are a consequence of a theorem of independent interest, which states that if H is a closed subgroup of a locally compact group G, and A is a closed subgroup of the center of H, such that A is normal in G, and (G/A, H/A) has relative Property (T), then (G, H^(1)) has relative Property (T), where H^(1) is the closure of the commutator subgroup of H. In fact, the assumption that A is in the center of H can be replaced with the weaker assumption that A is abelian and every H-invariant finite measure on the unitary dual of A is supported on the set of fixed points.

Journal of Group Theory, 2018

Let G be a finite group with the property that if a , b {a,b} are powers of δ 1 * {\delta_{1}^{*}} -commutators such that ( | a | , | b | ) = 1 {(|a|,|b|)=1} , then | a ⁢ b | = | a | ⁢ | b | {|ab|=|a||b|} . We show that γ ∞ ⁢ ( G ) {\gamma_{\infty}(G)} is nilpotent.

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