Tietze transformations

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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.

Group theory branch of mathematics that studies the algebraic properties of groups

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 axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right.

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. We then say G has presentation

Group (mathematics) set with an invertible, associative internal operation admitting a neutral element

In mathematics, a group is a set equipped with a binary operation which combines any two elements to form a third element in such a way that four conditions called group axioms are satisfied, namely closure, associativity, identity and invertibility. One of the most familiar examples of a group is the set of integers together with the addition operation, but groups are encountered in numerous areas within and outside mathematics, and help focusing on essential structural aspects, by detaching them from the concrete nature of the subject of the study.

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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.

Free group

In mathematics, the free groupFS over a given set S consists of all expressions that can be built from members of S, considering two expressions different unless their equality follows from the group axioms. The members of S are called generators of FS. An arbitrary group G is called free if it is isomorphic to FS for some subset S of G, that is, if there is a subset S of G such that every element of G can be written in one and only one way as a product of finitely many elements of S and their inverses.

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 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.

Elementary abelian group

In group theory, an elementary abelian group is an abelian group in which every nontrivial element has order p. The number p must be prime, and the elementary abelian groups are a particular kind of p-group. The case where p = 2, i.e., an elementary abelian 2-group, is sometimes called a Boolean group.

Examples

Let G = 〈 x,y | x3 = 1, y2 = 1, (xy)2 = 1 〉 be a presentation for the symmetric group 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).

Symmetric group automorphism group of a set; the group of bijections on a set, whose group operation is function composition

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 Sn defined over a finite set of n symbols consists of the permutation operations that can be performed on the n symbols. Since there are n! possible permutation operations that can be performed on a tuple composed of n symbols, it follows that the number of elements of the symmetric group Sn is n!.

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 = zy1 = zy and remove z = xy
G = 〈 y,z | (zy)3 = 1, y2 = 1, z2 = 1 〉 rule 4 - Remove the generator x

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References

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