On Restricting Subsets of Bases in Relatively Free Groups

2011/03/15 by Lucas Sabalka, Sabalka, Lucas, Dmytro Savchuk +1
Mathematics · #20E05 #20F16 #20F18 #FOS: Mathematics #Group Theory (math.GR) #math.GR #msc:20E05 #msc:20F16 #msc:20F18

paper · pdf · doi:10.48550/arxiv.1103.2992

9 pages; the proof of Theorem 4.1.(2) in the previous version contained an error. We prove a weaker statement in a newly added section Section 6

arxiv created 2013/02/03 · arxiv updated 2013/02/05

Abstract

Let G be a finitely generated free, free abelian of arbitrary exponent, free nilpotent, or free solvable group, or a free group in the variety A_mA_n, and let A = a₁,..., a_r be a basis for G. We prove that, in most cases, if S is a subset of a basis for G which may be expressed as a word in A without using elements from a_l+1,...,a_r, then S is a subset of a basis for the relatively free group on a₁,...,a_l.

Related

• Finding bases of uncountable free abelian groups is usually difficult
• Group bases for some solvable groups and semidirect products
• Free basis consisting of strictly small sections
• Forcing a Basis into ℵ₁-Free Groups
• The Strong Primitive Normal Basis Theorem