Elementary Nielsen transformation

Definition with symbols
Let $$F$$ be a free group. An elementary Nielsen transformation with respect to $$F$$ is a transformation that takes as input a finite (or countable) subset (or sequence) $$U$$ of $$F$$ and outputs a somewhat changed subset. The transformation could be of three types:


 * It replaces an element of $$U$$ by its inverse
 * It replaces an element of $$U$$ by its product with another element (the new element being placed on the right).
 * It deletes any element of $$U$$ which is equal to the identity element

When we are thinking in terms of a sequence rather than a subset, then we do not change the indexing of any of the other elements while performing the transformation.

A Nielsen transformation is a transformation which is obtained by composing one or more elementary Nielsen transformations.

Any Nielsen transformation on a freely generating set for $$U$$ gives rise to a corresponding automorphism, namely, the automorphism defined by sending the elements of the generating set to their images under the transformation. An automorphism arising from an elementary Nielsen transformation is termed an elementary Nielsen automorphism, while an automorphism arising from an arbitrary Nielsen transformation is termed a Nielsen automorphism. For finitely generated free groups, every automorphism is a Nielsen automorphism.