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

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

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).

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Datum:2004
1. Verfasser: Ercan, S.
Format: Artikel
Sprache:English
Veröffentlicht: Інститут прикладної математики і механіки НАН України 2004
Schriftenreihe:Algebra and Discrete Mathematics
Online Zugang:http://dspace.nbuv.gov.ua/handle/123456789/156415
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Zitieren:Torsion-free groups with every proper homomorphic image an N₁-group / S. Ercan // Algebra and Discrete Mathematics. — 2004. — Vol. 3, № 2. — С. 56–58. — Бібліогр.: 7 назв. — англ.

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spelling irk-123456789-1564152019-06-19T01:27:14Z Torsion-free groups with every proper homomorphic image an N₁-group Ercan, S. 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). 2004 Article Torsion-free groups with every proper homomorphic image an N₁-group / S. Ercan // Algebra and Discrete Mathematics. — 2004. — Vol. 3, № 2. — С. 56–58. — Бібліогр.: 7 назв. — англ. 1726-3255 2000 Mathematics Subject Classification: 20E15, 20F14. http://dspace.nbuv.gov.ua/handle/123456789/156415 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
language English
description 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).
format Article
author Ercan, S.
spellingShingle Ercan, S.
Torsion-free groups with every proper homomorphic image an N₁-group
Algebra and Discrete Mathematics
author_facet Ercan, S.
author_sort Ercan, S.
title Torsion-free groups with every proper homomorphic image an N₁-group
title_short Torsion-free groups with every proper homomorphic image an N₁-group
title_full Torsion-free groups with every proper homomorphic image an N₁-group
title_fullStr Torsion-free groups with every proper homomorphic image an N₁-group
title_full_unstemmed Torsion-free groups with every proper homomorphic image an N₁-group
title_sort torsion-free groups with every proper homomorphic image an n₁-group
publisher Інститут прикладної математики і механіки НАН України
publishDate 2004
url http://dspace.nbuv.gov.ua/handle/123456789/156415
citation_txt Torsion-free groups with every proper homomorphic image an N₁-group / S. Ercan // Algebra and Discrete Mathematics. — 2004. — Vol. 3, № 2. — С. 56–58. — Бібліогр.: 7 назв. — англ.
series Algebra and Discrete Mathematics
work_keys_str_mv AT ercans torsionfreegroupswitheveryproperhomomorphicimageann1group
first_indexed 2025-07-14T08:47:46Z
last_indexed 2025-07-14T08:47:46Z
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fulltext Jo u rn al A lg eb ra D is cr et e M at h . Algebra and Discrete Mathematics RESEARCH ARTICLE Number 2. (2004). pp. 56 – 58 c© Journal “Algebra and Discrete Mathematics” Torsion-free groups with every proper homomorphic image an N1-group Selami Ercan Communicated by D. Simson Abstract. 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). Introduction Let G be a group and let N be a normal subgroup of G. The factor group of G/N is said to be a proper factor-group if N 6= 1. Let G be a group, then G is called an N1-group if all subgroups of G are subnormal. N1-groups are considered by several authors and obtained remarkable results. 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 with non-trivial Fitting subgroup and every proper homomorphic image an N1-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 set IG(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 the isolator of H in G is a subgroup of G. If H is nilpotent of class c, where c 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, locally nilpotent groups, homomorphic image. Jo u rn al A lg eb ra D is cr et e M at h .S. Ercan 57 then 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 series H = 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 proper homomorphic images are N1-group. Then the Fitting subgroup of G is abelian. 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 N is non-abelian. Thus IG(N ′) is non-trivial and IG(N ′) / G. By hy- potehsis, G/IG(N ′) is an N1-group and G/IG(N ′) is a torsion-free group. G/IG(N ′) is nilpotent by [7 or 1]. Therefore G is nilpotent by Lemma 4.3.1 [3]. This is a contradiction. Thus N is an abelian group. Theorem. Let G be a torison-free locally nilpotent group with non-trivial Fitting 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 its Fitting subgroup. Then A is abelian by Lemma 1, so that also its isolator IG(A) is abelian. In particular, IG(A) ≤ CG(A) = A, so that IG(A) = A and G/A is torsion-free. Thus G/A is nilpotent. Let zA be a non-trivial element of Z(G/A). The map θ : a ∈ A −→ [a, z] is a G-endomorphism of A, and so C = kerθ = CA(z) is normal subgroup of 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/C is torsion-free, and so also nilpotent. Let yC be a non-trivial element of Z(G/C). If g is any element of G, then yg = yc with c ∈ C, and hence θ(y)g = θ(yg) = θ(yc) = θ(y), so that θ(y) ∈ Z(G) = 1. Thus y ∈ C = kerθ, and this contradiction proves the theorem. References [1] C. Casolo. Torsion-free groups in which every subgroup is subnormal, Rend. Circ. Mat. Palermo(2) -(2001), 321-324. Jo u rn al A lg eb ra D is cr et e M at h .58 Torsion-free groups... [2] P. Hall. The Edmontes notes on nilpotent groups. Queen Mary College Mathemat- ics Notes, London,(19690). [3] W. Möhres. Gruppen deren untergruppen alle subnormal sind. Würzburg Ph.D. thesis Aus Karlstadt, (1988). [4] W. Möhres. 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 and vols. 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 information S. Ercan Gazi Üniversitesi, Gazi Egitim Fakültesi, Matematik Egitimi, 06500 Teknikokullar, Ankara, Turkiye E-Mail: ercans@gazi.edu.tr Received by the editors: 09.04.2004 and final form in 24.05.2004.

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