In this article it is proved that a torsion-free

locally nilpotent groups with non-trivial Fitting subgroup and every

proper homomorphic image an N₁-group is an N₁-group(and so it

is nilpotent)

Ercan, S.

Algebra and Discrete Mathematics

Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)

Journal Algebra Discrete Math.Algebra and Discrete Mathematics RESEARCH ARTICLENumber 2. (2004). pp. 56 – 58c© Journal “Algebra and Discrete Mathematics”Torsion-free groups with every properhomomorphic image an N1-groupSelami ErcanCommunicated by D. SimsonAbstract. In this article it is proved that a torsion-freelocally nilpotent groups with non-trivial Fitting subgroup and everyproper homomorphic image an N1-group is an N1-group(and so itis nilpotent).IntroductionLet G be a group and let N be a normal subgroup of G. The factorgroup of G/N is said to be a proper factor-group if N 6= 1. Let G be agroup, then G is called an N1-group if all subgroups of G are subnormal.N1-groups are considered by several authors and obtained remarkableresults. If G is an torsion-free N1-group then G is soluble [4, (7)Satz],and nilpotent [7 or 1].In this article we consider locally nilpotent torsion-free groups withnon-trivial Fitting subgroup and every proper homomorphic image anN1-group. These groups are certain generalizations of N1-groups.Given a subgroup H of a group G, the isolator of H in G is the setIG(G) = {x ∈ G : xn ∈ H, 1 ≤ n for some natural number}If G is a locally nilpotent group and H is subgroup of G then theisolator of H in G is a subgroup of G. If H is nilpotent of class c, wherec is a natural number, then so is IG(H). If K is a normal subgroup of H,2000 Mathematics Subject Classification: 20E15, 20F14.Key words and phrases: all subgroups subnormal, torsion-free group, locallynilpotent groups, homomorphic image.Journal Algebra Discrete Math.S. Ercan 57then IG(K) is normal in IG(H)[2]. If H is a subnormal subgroup of G,we denote by d(G : H) the defect in G, i.e. the shortest length of a seriesH = H0 / H1 / . . . / Hd = G.The notations and definitions are standard and can be found in [5] and [6].Lemma 1. Let G be a locally nilpotent torsion-free group whose properhomomorphic images are N1-group. Then the Fitting subgroup of G isabelian.Proof. We show that every normal nilpotent subgroup of G is abelian.Let N be a non-trivial normal nilpotent subgroup of G such that Nis non-abelian. Thus IG(N′) is non-trivial and IG(N′) / G. By hy-potehsis, G/IG(N′) is anN1-group and G/IG(N′) is a torsion-free group.G/IG(N′) is nilpotent by [7 or 1]. Therefore G is nilpotent by Lemma4.3.1 [3]. This is a contradiction. Thus N is an abelian group.Theorem. Let G be a torison-free locally nilpotent group with non-trivialFitting subgroup. If all proper homomorphic images of G are N1-group,then G is an N1-group (and so it is nilpotent).Proof. Assume by contradiction that G is not N1-group, and let A be itsFitting subgroup. Then A is abelian by Lemma 1, so that also its isolatorIG(A) is abelian. In particular, IG(A) ≤ CG(A) = A, so that IG(A) = Aand G/A is torsion-free. Thus G/A is nilpotent. Let zA be a non-trivialelement of Z(G/A). The mapθ : a ∈ A −→ [a, z]is a G-endomorphism of A, and so C = kerθ = CA(z) is normal subgroupof G. On the other hand, if a is any non-trivial element of A, the subgroup< a, z > is nilpotent, so that A ∩ Z(< a, z >) 6= 1, and hence C 6= 1.Moreover, as [bn, z] = [b, z]n for all b ∈ A and n ∈ N , we have that G/Cis torsion-free, and so also nilpotent. Let yC be a non-trivial element ofZ(G/C). If g is any element of G, then yg = yc with c ∈ C, and henceθ(y)g = θ(yg) = θ(yc) = θ(y),so that θ(y) ∈ Z(G) = 1. Thus y ∈ C = kerθ, and this contradictionproves the theorem.References[1] C. Casolo. Torsion-free groups in which every subgroup is subnormal, Rend. Circ.Mat. Palermo(2) -(2001), 321-324.Journal Algebra Discrete Math.58 Torsion-free groups...[2] P. Hall. The Edmontes notes on nilpotent groups. Queen Mary College Mathemat-ics Notes, London,(19690).[3] W. Mo¨hres. Gruppen deren untergruppen alle subnormal sind. Wu¨rzburg Ph.D.thesis Aus Karlstadt, (1988).[4] W. Mo¨hres. Torsionfreie gruppen, deren untergruppen alle subnormal sind. Math.Ann. (1989), (284), 245-249.[5] D. J. S. Robinson, Finiteness condition and generalized soluble groups. vols. 1 andvols. 2 (Springer-Verlag, 1972).[6] D. J. S. Robinson, A course in the theory of groups. (Springer-Verlag, Heidelberg-Berlin-Newyork 1982.)[7] H. Smith, Torsion-free groups with all subgroups subnormal, Arch. Math.,(2001),(76),1-6.Contact informationS. Ercan Gazi U¨niversitesi, Gazi Egitim Faku¨ltesi,Matematik Egitimi, 06500 Teknikokullar,Ankara, TurkiyeE-Mail: ercans@gazi.edu.trReceived by the editors: 09.04.2004and final form in 24.05.2004.

Torsion-free groups with every proper homomorphic image an N₁-group

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Torsion-free groups with every proper homomorphic image an N₁-group

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