Nielsen transformation

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.