Automorphism Groups
In (Nielsen 1924), it is shown that the automorphism defined by the elementary Nielsen transformations generate the full automorphism group of a finitely generated free group. Nielsen, and later Neumann used these ideas to give finite presentations of the automorphism groups of free groups. This is also described in the textbook (Magnus, Karrass & Solitar 2004, p. 131, Th 3.2).
For a given generating set of a finite group (not necessarily free), not every automorphism is given by a Nielsen transformation, but for every automorphism, there is a generating set where the automorphism is given by a Nielsen transformation, (Rapaport 1959).
Read more about this topic: Nielsen Transformation, Applications
Famous quotes containing the word groups:
“In properly organized groups no faith is required; what is required is simply a little trust and even that only for a little while, for the sooner a man begins to verify all he hears the better it is for him.”
—George Gurdjieff (c. 1877–1949)