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 ${\displaystyle F_{n}}$, the corresponding set of elementary Nielsen transformations forms a finite generating set of ${\displaystyle \mathrm {Aut} (F_{n})}$. This system of generators is analogous to elementary matrices for ${\displaystyle GL_{n}(\mathbb {Z} )}$ 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.

Let ${\textstyle F_{n}}$ be a finitely generated free group of rank ${\textstyle n}$. An elementary Nielsen transformation maps an ordered basis ${\textstyle [x_{1},\ldots ,x_{n}]}$ to a new basis ${\textstyle [y_{1},\ldots ,y_{n}]}$ by one of the following operations:

A Nielsen transformation is a finite composition of elementary Nielsen transformations. Since automorphisms of ${\displaystyle F_{n}}$ are determined by the image of a basis, the elementary Nielsen transformations correspond to a finite subset of the automorphism group ${\textstyle \mathrm {Aut} (F_{n})}$, which is in fact a generating set (see below). Hence, Nielsen transformation can alternatively be defined simply as the action of an automorphism of ${\textstyle F_{n}}$ on bases.

Elementary Nielsen transformations are the analogues of the elementary row operations. Transformations of the first kind are analogous to row permutations. Transformations of the second kind correspond to scaling a row by an invertible scalar. Transformations of the third kind correspond to row additions (transvections).

Since the finite permutation group ${\displaystyle S_{n}}$ is generated by transpositions, one sees from the chain of elementary Nielsen transformations of type 2 and 3:$${\displaystyle (x_{i},x_{i+1})\mapsto {(x_{i},x_{i+1}^{-1})}\mapsto {{(x_{i}x_{i+1}^{-1}},x_{i+1}^{-1})}\mapsto ({{x_{i+1}x_{i}^{-1}},x_{i+1}^{-1}})\mapsto {({x_{i+1}x_{i}^{-1}},x_{i}^{-1})}\mapsto (x_{i+1}x_{i}^{-1},x_{i})\mapsto (x_{i+1},x_{i})}$$that elementary Nielsen transformations of type 2 and 3 are in fact enough to generate all Nielsen transformations.

Using the two generators ${\textstyle (12)}$ and ${\textstyle (1\ldots n)}$ of ${\textstyle S_{n}}$, one can alternatively restrict attention to only four operations: