On the Nilpotency Class of a Generalized 3-Abelian Group

A group G is called 3-abelian if the map \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${x \mapsto x^{3}}$$\end{document} is an endomorphism of G and it is called generalized 3-abelian, if there exist elements \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${c_{1}, c_{2}, c_{3} \in G}$$\end{document} such that the map \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varphi : x \longmapsto {x^{c_{1}} x^{c_{2}} x^{c_{3}}}}$$\end{document} 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.


Introduction and Results
Let n ≥ 2 be an integer. A group G is called n-abelian whenever (xy) n = x n y n for all x, y ∈ G; or equivalently the map x → x n is an endomorphism of G. Levi [7] has proved that a group G is 3-abelian if and only if it is 2-Engel and the exponent of its derived subgroup [G, G] divides 3. Trotter [10] has proved that a 3-abelian group G is abelian whenever the map x → x 3 is an automorphism of G. A group G is called generalized n-abelian whenever there exist elements c 1 , . . . , c n ∈ G such that the map x → x c1 ... x cn is an endomorphism of G. The class of generalized n-abelian groups is closed under the formation images, and finite direct products. Obviously, every n-abelian group is generalized n-abelian and it is easy to see that every generalized 2-abelian group is abelian. It is clear that by conjugating we may assume one of c 1 , . . . , c n to be the trivial element.
Abdollahi, Daoud and Endimioni [1, Theorem 3.1] have proved that a generalized 3-abelian group G is nilpotent of class at most 10, and abelian Theorem 1.2. Let G be a generalized 3-abelian group admitting an endomorphism of the form ii) The subgroup Imφ is abelian. In particular, if φ is injective or surjective, G is abelian.
Let m = 0 be an integer. Baer [2] introduced the m-center of a group G as follows: The set Z(G, m) is a characteristic subgroup of G for any non-zero integer m. L.-C. Kappe and M. L. Newel [5] proved that Thus only one of the m-commutativity conditions suffices to define the mcenter Z(G, m). If m is a positive integer, the upper m-central series Z i (G, m) is defined inductively as the following: We then get an ascending series.

Proofs
Notations used in this paper are standard. For a group G and the elements x 1 , x 2 , ..., x n , x, y ∈ G, the commutators [x 1 , x 2 , ..., x n ] and [x n , y] are defined inductively by the rules: For a given integer i ≥ 1, we denote by [G, G], ζ i (G) and γ 3 (G) respectively the derived subgroup, the ith-center and the third term of the lower central series of G. We denote by H ≤ G if H is a subgroup of G. To prove Theorems 1.1, 1.2 and 1.3 we need the following lemmas. Lemma 2.1. Let G be a generalized n-abelian group admitting an endomorphism of the form ψ : x → x c1 · · · x cn where c 1 , . . . , c n ∈ G. Then, G is n-abelian whenever c 1 , . . . , c n ∈ ζ 2 (G).
whence (xy) n = x n y n .

Lemma 2.2.
Let G be a metabelian group, x, y ∈ G and u, v ∈ [G, G]. Then, for any integer n ≥ 1, the following assertions hold.
Proof. (a) is easy to prove as [G, G] is abelian. For the proofs of (b) and (c) see [4].
Proof of Theorem 1.1. i) Let G be a generalized 3-abelian group with the given endomorphism φ defined by x φ = x a xx b for all x ∈ G, where a, b ∈ G are fixed. To prove that G is nilpotent of class three, it is enough to show that every 4-generated subgroup g 1 , g 2 , g 3 , g 4 of G is nilpotent of class at most 3. Now H = g 1 , g 2 , g 3 , g 4 , a, b is clearly invariant under the action of φ and is thus a generalized 3-abelian group as well. We can thus replace G by H and it suffices then to show that H is nilpotent of class at most 3. By [1, Théorème 3.1], H is nilpotent. Now one can use nq package of Werner Nickel [8] implemented in GAP [9] and MAGMA [3] to find the nilpotency class Mediterr. J. Math.
of H. The package nq has the capability of computing the largest nilpotent quotient (if it exists) of a finitely generated group with finitely many identical relations and finitely many relations. For example, if we want to construct the largest nilpotent quotient of a group G with the following presentation . . . , x n , y 1 , . . . , y k ) = 1 ,   where r 1 , . . . , r m are relations on x 1 , . . . , x n and w(x 1 , . . . , x n , y 1 , . . . , y k ) = 1 is an identical relation in the group x 1 , . . . , x n , one may apply the following code to use the package nq in GAP: LoadPackage("nq"); #nq package of Werner Nickel Note that we need to construct the free group of rank n+k because as well as the n generators for G we also have an identical relation with k free variables. Note that the function NilpotentQuotient(L) attempts to compute the largest nilpotent quotient of L and it will terminate only if L has a largest nilpotent quotient. Note that our identical relation is (xy) φ = x φ y φ for all x, y ∈ G, which can be written as follows: ii) Let x, y ∈ G. From part (i) and Theorem 1.1, we know that G is nilpotent of class at most 3 and γ 3 (G) 3 = {1}. Therefore Using furthermore the facts from part (i) and Theorem 1.