In
group theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
, Tietze transformations are used to transform a given
presentation of a group
In mathematics, a presentation is one method of specifying a group. A presentation of a group ''G'' comprises a set ''S'' of generators—so that every element of the group can be written as a product of powers of some of these generators—and ...
into another, often simpler presentation of the same
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 iden ...
. These transformations are named after
Heinrich Franz Friedrich Tietze
Heinrich Franz Friedrich Tietze (August 31, 1880 – February 17, 1964) was an Austrian mathematician, famous for the Tietze extension theorem on functions from topological spaces to the real numbers. He also developed the Tietze transformat ...
who introduced them in a paper in 1908.
A presentation is in terms of ''generators'' and ''relations''; formally speaking the presentation is a pair of a set of named generators, and a set of words in the
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' ...
on the generators that are taken to be the relations. Tietze transformations are built up of elementary steps, each of which individually rather evidently takes the presentation to a presentation of an
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
group. These elementary steps may operate on generators or relations, and are of four kinds.
Adding a relation
If a relation can be derived from the existing relations then it may be added to the presentation without changing the group. Let G=〈 x , x
3=1 〉 be a finite presentation for the cyclic group of order 3. Multiplying x
3=1 on both sides by x
3 we get x
6 = x
3 = 1 so x
6 = 1 is derivable from x
3=1. Hence G=〈 x , x
3=1, x
6=1 〉 is another presentation for the same group.
Removing a relation
If a relation in a presentation can be derived from the other relations then it can be removed from the presentation without affecting the group. In ''G'' = 〈 ''x'' , ''x''
3 = 1, ''x''
6 = 1 〉 the relation ''x''
6 = 1 can be derived from ''x''
3 = 1 so it can be safely removed. Note, however, that if ''x''
3 = 1 is removed from the presentation the group ''G'' = 〈 ''x'' , ''x''
6 = 1 〉 defines the cyclic group of order 6 and does not define the same group. Care must be taken to show that any relations that are removed are consequences of the other relations.
Adding a generator
Given a presentation it is possible to add a new generator that is expressed as a word in the original generators. Starting with ''G'' = 〈 ''x'' , ''x''
3 = 1 〉 and letting ''y'' = ''x''
2 the new presentation ''G'' = 〈 ''x'',''y'' , ''x''
3 = 1, ''y'' = ''x''
2 〉 defines the same group.
Removing a generator
If a relation can be formed where one of the generators is a word in the other generators then that generator may be removed. In order to do this it is necessary to replace all occurrences of the removed generator with its equivalent word. The presentation for the
elementary abelian group
In mathematics, specifically in group theory, an elementary abelian group (or elementary abelian ''p''-group) is an abelian group in which every nontrivial element has order ''p''. The number ''p'' must be prime, and the elementary abelian group ...
of order 4, G=〈 x,y,z , x = yz, y
2=1, z
2=1, x=x
−1 〉 can be replaced by ''G'' = 〈 ''y'',''z'' , ''y''
2 = 1, ''z''
2 = 1, (''yz'') = (''yz'')
−1 〉 by removing ''x''.
Examples
Let ''G'' = 〈 ''x'',''y'' , ''x''
3 = 1, ''y''
2 = 1, (''xy'')
2 = 1 〉 be a presentation for the
symmetric group
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group \m ...
of degree three. The generator ''x'' corresponds to the permutation (1,2,3) and ''y'' to (2,3). Through Tietze transformations this presentation can be converted to ''G'' = 〈 ''y'', ''z'' , (''zy'')
3 = 1, ''y''
2 = 1, ''z''
2 = 1 〉, where z corresponds to (1,2).
See also
*
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 ...
*
Andrews-Curtis Conjecture
References
*
Roger C. Lyndon
Roger Conant Lyndon (December 18, 1917 – June 8, 1988) was an American mathematician, for many years a professor at the University of Michigan.. He is known for Lyndon words, the Curtis–Hedlund–Lyndon theorem, Craig–Lyndon interpolation a ...
,
Paul E. Schupp
Paul Eugene Schupp (born March 12, 1937, died January 24, 2022) was a professor emeritus of mathematics at the University of Illinois at Urbana Champaign. He is known for his contributions to geometric group theory, computational complexity and th ...
, ''Combinatorial Group Theory'', Springer, 2001. {{isbn, 3-540-41158-5.
Combinatorial group theory