Premium
Presentations of Free Abelian‐by‐(Nilpotent of Class 2) Groups
Author(s) -
Evans Martin J.
Publication year - 1998
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/s0024609397003676
Subject(s) - mathematics , combinatorics , abelian group , epimorphism , free product , rank (graph theory) , free group , nilpotent , nilpotent group , closure (psychology) , triangular matrix , group (periodic table) , pure mathematics , law , chemistry , invertible matrix , organic chemistry , political science
Let F n , M n and U n denote the free group of rank n , the free metabelian group of rank n , and the free abelian‐by‐(nilpotent of class 2) group of rank n , respectively. Thus M n ≅ F n / F ″ n and U n ≅ F n /[γ 3 ( F n ), γ 3 ( F n )], where γ 3 ( F n ) = [ F ′ n , F n ], the third term of the lower central series of F n . Consider an arbitrary epimorphism θ: F n ↠ F k , where k < n . A routine argument (see [ 10 , Theorem 3.3]) using the Nielsen reduction process shows that there exists a free basis { y 1 , y 2 , …, y n } of F n such that ker θ is the normal closure in F n of { y 1 , y 2 , …, y n−k }. A similar result has been obtained for free metabelian groups by C. K. Gupta, N. D. Gupta and G. A. Noskov [ 5 ]. Indeed, it follows immediately from [ 5 , Theorem 3.1] that if k < n and θ: M n ↠ M k is an epimorphism, then there exists a free basis { m 1 , m 2 , …, m n } of M n such that ker θ is the normal closure in M n of { m 1 , m 2 , …, m n−k }. 1991 Mathematics Subject Classification 20F05, 20E10.