Definitions
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 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. Two generating sets are called Nielsen equivalent if there is a Nielsen transformation taking one to the other. If the generating sets have the same size, then it suffices to consider compositions of regular, elementary Nielsen transformations.
Read more about this topic: Nielsen Transformation
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