In mathematics, especially in the area of modern algebra known as combinatorial group theory, Nielsen transformations are certain automorphisms of a free group which are a non-commutative analogue of row reduction and one of the main tools used in studying free groups (Fine, Rosenberger & Stille 1995). Given a finite basis of a free group F n {\displaystyle F_{n}} , the corresponding set of elementary Nielsen transformations forms a finite generating set of A u t ( F n ) {\displaystyle \mathrm {Aut} (F_{n})} . This system of generators is analogous to elementary matrices for G L n ( Z ) {\displaystyle GL_{n}(\mathbb {Z} )} and Dehn twists for mapping class groups of closed surfaces. Nielsen transformations were introduced in (Nielsen 1921) to prove that every subgroup of a free group is free (the Nielsen–Schreier theorem). They are now used in a variety of mathematics, including computational group theory, k-theory, and knot theory.
Definitions
Free groups Let F n {\textstyle F_{n}} be a finitely generated free group of rank n {\textstyle n} . An elementary Nielsen transformation maps an ordered basis [ x 1 , … , x n ] {\textstyle [x_{1},\ldots ,x_{n}]} to a new basis [ y 1 , … , y n ] {\textstyle [y_{1},\ldots ,y_{n}]} by one of the following operations:
Permute the x i {\textstyle x_{i}} s by some permutation σ ∈ S n {\textstyle \sigma \in S_{n}} , i.e. [ y 1 , … , y n ] = [ x σ ( 1 ) , … , x σ ( n ) ] {\textstyle [y_{1},\ldots ,y_{n}]=[x_{\sigma (1)},\ldots ,x_{\sigma (n)}]}
Invert some x i {\textstyle x_{i}} , i.e. [ y 1 , … , y n ] = [ x 1 , … , x i − 1 , … , x n ] {\textstyle [y_{1},\ldots ,y_{n}]=[x_{1},\ldots ,x_{i}^{-1},\ldots ,x_{n}]}
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