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On Restricting Subsets of Bases in Relatively Free Groups

By Lucas Sabalka and Dmytro Savchuk


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 = {a_1,..., 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 {a_1,...,a_l}.Comment: 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

Topics: Mathematics - Group Theory, 20E05, 20F18, 20F16
Year: 2013
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