Tietze transformations
Encyclopedia
In group theory
, Tietze transformations are used to transform a given presentation of a group
into another, often simpler presentation of the same group
. These transformations are named after Heinrich Franz Friedrich Tietze
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
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 group. These elementary steps may operate on generators or relations, and are of four kinds.
of order 4, G=〈 x,y,z | x = yz, y2=1, z2=1, x=x-1 〉 can be replaced by G = 〈 y,z | y2 = 1, z2 = 1, (yz) = (yz)−1 〉 by removing x.
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, y2 = 1, z2 = 1 〉, where z corresponds to (1,2).
Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
, Tietze transformations are used to transform a given presentation of a group
Presentation of a group
In mathematics, one method of defining a group is by a presentation. One specifies 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 a set R of relations among those generators...
into another, often simpler presentation of the same group
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...
. These transformations are named after Heinrich Franz Friedrich Tietze
Heinrich Franz Friedrich Tietze
Heinrich Franz Friedrich Tietze was an Austrian mathematician, famous for the Tietze extension theorem. He also developed the Tietze transformations for group presentations, and was the first to pose the group isomorphism problem.He was born in Schleinz, Austria and died in Munich,...
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
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...
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 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 | x3=1 〉 be a finite presentation for the cyclic group of order 3. Multiplying x3=1 on both sides by x3 we get x6 = x3 = 1 so x6 = 1 is derivable from x3=1. Hence G=〈 x | x3=1, x6=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 | x3 = 1, x6 = 1 〉 the relation x6 = 1 can be derived from x3 = 1 so it can be safely removed. Note, however, that if x3 = 1 is removed from the presentation the group G = 〈 x | x6 = 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 | x3 = 1 〉 and letting y = x2 the new presentation G = 〈 x,y | x3 = 1, y = x2 〉 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 groupElementary Abelian group
In group theory, an elementary abelian group is a finite abelian group, where every nontrivial element has order p, where p is a prime; in particular it is a p-group....
of order 4, G=〈 x,y,z | x = yz, y2=1, z2=1, x=x-1 〉 can be replaced by G = 〈 y,z | y2 = 1, z2 = 1, (yz) = (yz)−1 〉 by removing x.
Examples
Let G = 〈 x,y | x3 = 1, y2 = 1, (xy)2 = 1 〉 be a presentation for the symmetric groupSymmetric group
In mathematics, the symmetric group Sn on a finite set of n symbols is the group whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself...
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, y2 = 1, z2 = 1 〉, where z corresponds to (1,2).
| G = 〈 x,y | | x3 = 1, y2 = 1, (xy)2 = 1 〉 | (start) |
| G = 〈 x,y,z | | x3 = 1, y2 = 1, (xy)2 = 1, z = xy 〉 | rule 3 — Add the generator z |
| G = 〈 x,y,z | | x3 = 1, y2 = 1, (xy)2 = 1, x = zy 〉 | rules 1 and 2 — Add x = zy−1 = zy and remove z = xy |
| G = 〈 y,z | | (zy)3 = 1, y2 = 1, z2 = 1 〉 | rule 4 - Remove the generator x |
See also
- Nielsen TransformationNielsen transformationIn 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,...
- Andrews-Curtis ConjectureAndrews–Curtis conjectureIn mathematics, the Andrews–Curtis conjecture states that every balanced presentation of the trivial group can be transformed into a trivial presentation by a sequence of Nielsen transformations on the relators together with conjugations of relators, named after James J. Andrews and Morton L....