Displaying similar documents to “A nilpotency condition for finitely generated soluble groups”

Infinite locally soluble k -Engel groups

Lucia Serena Spiezia (1992)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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In this paper we deal with the class E k * of groups G for which whenever we choose two infinite subsets X , Y there exist two elements x ∈ X , y ∈ Y such that x , y , … , y ︸ k = 1 . We prove that an infinite finitely generated soluble group in the class E k * is in the class E k of k -Engel groups. Furthermore, with k = 2 , we show that if G ∈ E 2 * is infinite locally soluble or hyperabelian group then G ∈ E 2 .

On absolutely-nilpotent of class k groups

Patrizia Longobardi, Trueman MacHenry, Mercede Maj, James Wiegold (1995)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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A group G in a variety V is said to be absolutely- V , and we write G ∈ A ⁢ V , if central extensions by G are again in V . Absolutely-abelian groups have been classified by F. R. Beyl. In this paper we concentrate upon the class A ⁢ N k of absolutely-nilpotent of class k groups. We prove some closure properties of the class A ⁢ N k and we show that every nilpotent of class k group can be embedded in an A ⁢ N k -gvoup. We describe all metacyclic A ⁢ N k -groups and we characterize 2 -generator and infinite 3 -generator A ⁢ N 2 -groups....

Groups whose proper subgroups are locally finite-by-nilpotent

Amel Dilmi (2007)

Annales mathématiques Blaise Pascal

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If 𝒳 is a class of groups, then a group G is said to be minimal non 𝒳 -group if all its proper subgroups are in the class 𝒳 , but G itself is not an 𝒳 -group. The main result of this note is that if c > 0 is an integer and if G is a minimal non ( ℒℱ ) 𝒩 (respectively, ( ℒℱ ) 𝒩 c )-group, then G is a finitely generated perfect group which has no non-trivial finite factor and such that G / F r a t ( G ) is an infinite simple group; where 𝒩 (respectively, 𝒩 c , ℒℱ ) denotes the class of nilpotent (respectively, nilpotent of class...

Hurewicz-Serre theorem in extension theory

M. Cencelj, J. Dydak, A. Mitra, A. Vavpetič (2008)

Fundamenta Mathematicae

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The paper is devoted to generalizations of the Cencelj-Dranishnikov theorems relating extension properties of nilpotent CW complexes to their homology groups. Here are the main results of the paper: Theorem 0.1. Let L be a nilpotent CW complex and F the homotopy fiber of the inclusion i of L into its infinite symmetric product SP(L). If X is a metrizable space such that X τ K ( H k ( L ) , k ) for all k ≥ 1, then X τ K ( π k ( F ) , k ) and X τ K ( π k ( L ) , k ) for all k ≥ . Theorem 0.2. Let X be a metrizable space such that dim(X) < ∞ or X ∈...

Parallelepipeds, nilpotent groups and Gowers norms

Bernard Host, Bryna Kra (2008)

Bulletin de la Société Mathématique de France

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In his proof of Szemerédi’s Theorem, Gowers introduced certain norms that are defined on a parallelepiped structure. A natural question is on which sets a parallelepiped structure (and thus a Gowers norm) can be defined. We focus on dimensions 2 and 3 and show when this possible, and describe a correspondence between the parallelepiped structures and nilpotent groups.

The Hughes subgroup

Robert Bryce (1994)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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Let G be a group and p a prime. The subgroup generated by the elements of order different from p is called the Hughes subgroup for exponent p . Hughes [3] made the following conjecture: if H p ⁢ G is non-trivial, its index in G is at most p . There are many articles that treat this problem. In the present Note we examine those of Strauss and Szekeres [9], which treats the case p = 3 and G arbitrary, and that of Hogan and Kappe [2] concerning the case when G is metabelian, and p arbitrary. A common...

Convergence of formal solutions of first order singular partial differential equations of nilpotent type

Masatake Miyake, Akira Shirai (2012)

Banach Center Publications

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Let (x,y,z) ∈ ℂ³. In this paper we shall study the solvability of singular first order partial differential equations of nilpotent type by the following typical example: P u ( x , y , z ) : = ( y ∂ x - z ∂ y ) u ( x , y , z ) = f ( x , y , z ) ∈ x , y , z , where P = y ∂ x - z ∂ y : x , y , z → x , y , z . For this equation, our aim is to characterize the solvability on x , y , z by using the Im P, Coker P and Ker P, and we give the exact forms of these sets.