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On the Bergman Property for the Automorphism Groups of Relatively Free Groups
Author(s) -
Tolstykh Vladimir
Publication year - 2006
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610706023052
Subject(s) - automorphism , mathematics , inner automorphism , group (periodic table) , free product , free group , outer automorphism group , property (philosophy) , nilpotent , rank (graph theory) , countable set , p group , automorphism group , combinatorics , pure mathematics , physics , philosophy , epistemology , quantum mechanics
A group G is said to have the Bergman property (the property of uniformity of finite width) if given any generating X with X = X −1 of G , we have that G = X k for some natural k , that is, every element of G is a product of at most k elements of X . We prove that the automorphism group Aut( N ) of any infinitely generated free nilpotent group N has the Bergman property. Also, we obtain a partial answer to a question posed by Bergman by establishing that the automorphism group of a free group of countably infinite rank is a group of uniformly finite width.