Nielsen-Transformation

In mathematics Nielsen transformations are an important tool in combinatorial group theory, they are named after the mathematician Jakob Nielsen.

Definition

Is a group and a n-tuple of elements geordenetes from G. An elementary Nielsen transform is one of the following three types of substitutions:

• For a replace by.
• For two and Swap.
• For two replaced by.

Nielsen a transformation is a sequence of a finite number of elementary transformations Nielsen. Two ordered tuple glad Nielsen equivalent if they emerge by a Nielsen transformation apart.

Applications

Systems of generators of free groups

Be the free group with n generators. Then every minimal generating set has n elements and an n-tuple is exactly then a system of generators if the parent tuple and Nielsen equivalent.

Systems of generators of surface groups

Be the surface group of genus g then every minimal system of generators has 2g elements and a 2g - tuple is then precisely a system of generators if the parent tuple and Nielsen equivalent.