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 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 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''− ...

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 or morphism 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 the ...

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'' , ''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 is an abelian group in which all elements other than the identity have the same order. This common order must be a prime number, and the elementary abelian groups in whic ...

of order 4, G=〈 x,y,z , x = yz, y²=1, z²=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 grou ...

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).

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 ...

Paul E. Schupp Paul Eugene Schupp (March 12, 1937 – January 24, 2022) was an American-born British professor emeritus of mathematics at the University of Illinois at Urbana Champaign. He is known for his contributions to geometric group theory, computational c ...