In mathematical group theory, the automorphism group of a free group is a discrete
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
of
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 ...
s of a
free group
In mathematics, the free group ''F'S'' over a given set ''S'' consists of all words that can be built from members of ''S'', considering two words to be different unless their equality follows from the group axioms (e.g. ''st'' = ''suu''−1' ...
. The quotient by the inner automorphisms is the
outer automorphism group of a free group
In mathematics, Out(''Fn'') is the outer automorphism group of a free group on ''n'' generators. These groups play an important role in geometric group theory.
Outer space
Out(''Fn'') acts geometrically on a cell complex known as Culler– V ...
, which is similar in some ways to the
mapping class group of a surface In mathematics, and more precisely in topology, the mapping class group of a surface, sometimes called the modular group or Teichmüller modular group, is the group of homeomorphisms of the surface viewed up to continuous (in the compact-open topolo ...
.
Presentation
showed that the automorphisms defined by the elementary
Nielsen 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 ...
s generate the full automorphism group of a finitely generated free group. Nielsen, and later
Bernhard Neumann
Bernhard Hermann Neumann (15 October 1909 – 21 October 2002) was a German-born British-Australian mathematician, who was a leader in the study of group theory.
Early life and education
After gaining a D.Phil. from Friedrich-Wilhelms Universit ...
used these ideas to give
finite presentations of the
automorphism group
In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the g ...
s of free groups. This is also described in .
The automorphism group of the free group with ordered basis
''x''1, …, ''x''''n'' is generated by the following 4 elementary Nielsen transformations:
* Switch ''x''
1 and ''x''
2
* Cyclically permute ''x''
1, ''x''
2, …, ''x''
''n'', to ''x''
2, …, ''x''
''n'', ''x''
1.
* Replace ''x''
1 with ''x''
1−1
* Replace ''x''
1 with ''x''
1·''x''
2
These transformations are the analogues of the
elementary row operations In mathematics, an elementary matrix is a matrix which differs from the identity matrix by one single elementary row operation. The elementary matrices generate the general linear group GL''n''(F) when F is a field. Left multiplication (pre-multi ...
. 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 .
See Also
*
Out(Fn)
In mathematics, Out(''Fn'') is the outer automorphism group of a free group on ''n'' generators. These groups play an important role in geometric group theory.
Outer space
Out(''Fn'') acts geometrically on a cell complex known as Culler–Vog ...
References
*
*
*
*{{Citation , last1=Vogtmann , first1=Karen , authorlink=Karen Vogtmann , department=Proceedings of the Conference on Geometric and Combinatorial Group Theory, Part I (Haifa, 2000) , url=http://www.math.cornell.edu/~vogtmann/papers/Autosurvey/autosurvey.pdf , doi=10.1023/A:1020973910646 , mr=1950871 , year=2002 , journal=Geometriae Dedicata , issn=0046-5755 , volume=94 , title=Automorphisms of free groups and outer space , pages=1–31
Combinatorial group theory