Concept
In mathematics, especially in the area of abstract algebra known as combinatorial group theory, Nielsen transformations, named after Jakob Nielsen, 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, . They were introduced in to prove that every subgroup of a free group is free (the Nielsen–Schreier theorem), but are now used in a variety of mathematics, including computational group theory, k-theory, and knot theory. The textbook devotes all of chapter 3 to Nielsen transformations. One of the simplest definitions of a Nielsen transformation is an automorphism of a free group, but this was not their original definition. The following gives a more constructive definition. A Nielsen transformation on a finitely generated free group with ordered basis [ x1, ..., xn ] can be factored into elementary Nielsen transformations of the following sorts: Switch x1 and x2 Cyclically permute x1, x2, ..., xn, to x2, ..., xn, x1. Replace x1 with x1−1 Replace x1 with x1·x2 These transformations are the analogues of the elementary row operations. Transformations of the first two kinds are analogous to row swaps, and cyclic row permutations. Transformations of the third kind correspond to scaling a row by an invertible scalar. Transformations of the fourth kind correspond to row additions. Transformations of the first two types suffice to permute the generators in any order, so the third type may be applied to any of the generators, and the fourth type to any pair of generators. When dealing with groups that are not free, one instead applies these transformations to finite ordered subsets of a group. In this situation, compositions of the elementary transformations are called regular. If one allows removing elements of the subset that are the identity element, then the transformation is called singular. The image under a Nielsen transformation (elementary or not, regular or not) of a generating set of a group G is also a generating set of G.