Some Results on Free Groups in Combinatorial Group Theory
Lee, Donghi
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https://hdl.handle.net/2142/86777
Description
Title
Some Results on Free Groups in Combinatorial Group Theory
Author(s)
Lee, Donghi
Issue Date
2001
Doctoral Committee Chair(s)
Sergei v. Ivanov
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Mathematics
Language
eng
Abstract
Let Fm be a free group of a finite rank m ≥ 2. We prove that there exist two elements u 1, u2 ∈ Fm such that every endomorphism y of Fm with non-cyclic image is completely determined by y (u1), y (u2). We obtain this result as a corollary of the construction of a C-test word vn( x1,...,xn), for each n ≥ 2, with the additional property that vn (X1,...,Xn) = 1 if and only if the subgroup 〈X1,..., Xn〉 of Fm generated by X1,...,Xn is cyclic. We also prove that every primitivity preserving endomorphism of Fm with m ≥ 3 is an automorphism.
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