Automorphism group of a free group
Encyclopedia
In mathematical group theory, the automorphism group of a free group is a discrete group
of automorphism
s of a free group
. The quotient by the inner automorphisms is the outer automorphism group of a free group, which is similar in some ways to the mapping class group of a surface.
s generates the full automorphism group of a finitely generated free group. Nielsen, and later Bernhard Neumann
used these ideas to give finite presentations of the automorphism groups of free groups. This is also described in .
The automorphism group of the free group with ordered basis [ x1, …, xn ] is generated by the following 4 elementary Nielsen transformations:
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.
Nielsen gave a rather complicated finite presentation using these generators, described in .
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
of automorphism
Automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called the automorphism...
s of a free group
Free group
In mathematics, a group G is called free if there is a subset S of G such that any element of G can be written in one and only one way as a product of finitely many elements of S and their inverses...
. The quotient by the inner automorphisms is the outer automorphism group of a free group, which is similar in some ways to the mapping class group of a surface.
Presentation
showed that the automorphism defined by the elementary Nielsen transformationNielsen transformation
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,...
s generates the full automorphism group of a finitely generated free group. Nielsen, and later Bernhard Neumann
Bernhard Neumann
Bernhard Hermann Neumann AC FRS was a German-born British mathematician who was one of the leading figures in group theory, greatly influencing the direction of the subject....
used these ideas to give finite presentations of the automorphism groups of free groups. This is also described in .
The automorphism group of the free group with ordered basis [ x1, …, xn ] is generated by the following 4 elementary Nielsen transformations:
- 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
Elementary row operations
In mathematics, an elementary matrix is a simple matrix which differs from the identity matrix by one single elementary row operation. The elementary matrices generate the general linear group of invertible matrices...
. 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.
Nielsen gave a rather complicated finite presentation using these generators, described in .