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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.

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

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.

- Adding a relation
- Removing a relation
- Adding a generator
- Removing a generator
- Examples
- See also
- References

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.

In mathematics, the **free group***F*_{S} 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 *F*_{S}. An arbitrary group *G* is called **free** if it is isomorphic to *F*_{S} 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.

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.

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.

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.

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, 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*.

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**.

Let *G* = 〈 *x*,*y* | *x*^{3} = 1, *y*^{2} = 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, *y*^{2} = 1, *z*^{2} = 1 〉, where z corresponds to (1,2).

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 **S _{n}** defined over a finite set of

G = 〈 x,y | x^{3} = 1, y^{2} = 1, (xy)^{2} = 1 〉 | (start) |

G = 〈 x,y,z| x^{3} = 1, y^{2} = 1, (xy)^{2} = 1, z = xy 〉 | rule 3 — Add the generator z |

G = 〈 x,y,z | x^{3} = 1, y^{2} = 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, y^{2} = 1, z^{2} = 1 〉 | rule 4 - Remove the generator x |

In mathematics, an **equivalence relation** is a binary relation that is reflexive, symmetric and transitive. The relation "is equal to" is the canonical example of an equivalence relation, where for any objects a, b, and c:

In physics, the **Lorentz transformations** are a one-parameter family of linear transformations from a coordinate frame in space time to another frame that moves at a constant velocity, the parameter, within the former. The transformations are named after the Dutch physicist Hendrik Lorentz. The respective inverse transformation is then parametrized by the negative of this velocity.

In algebra, a **cyclic group** or **monogenous group** is a group that is generated by a single element. That is, it consists of a set of elements with a single invertible associative operation, and it contains an element *g* such that every other element of the group may be obtained by repeatedly applying the group operation or its inverse to *g*. Each element can be written as a power of *g* in multiplicative notation, or as a multiple of *g* in additive notation. This element *g* is called a *generator* of the group.

In abstract algebra, a **generating set of a group** is a subset such that every element of the group can be expressed as the combination of finitely many elements of the subset and their inverses.

In mathematics, **function composition** is an operation that takes two functions *f* and *g* and produces a function *h* such that *h*(*x*) = *g*(*f* ). In this operation, the function *g* is applied to the result of applying the function *f* to *x*. That is, the functions *f* : *X* → *Y* and *g* : *Y* → *Z* are **composed** to yield a function that maps *x* in *X* to *g*(*f* ) in *Z*.

In mathematics, a **Cayley graph**, also known as a **Cayley colour graph**, **Cayley diagram**, **group diagram**, or **colour group** is a graph that encodes the abstract structure of a group. Its definition is suggested by Cayley's theorem and uses a specified, usually finite, set of generators for the group. It is a central tool in combinatorial and geometric group theory.

In mathematics, specifically group theory, the **free product** is an operation that takes two groups *G* and *H* and constructs a new group *G* ∗ *H*. The result contains both *G* and *H* as subgroups, is generated by the elements of these subgroups, and is the “most general” group having these properties. Unless one of the groups *G* and *H* is trivial, the free product is always infinite. The construction of a free product is similar in spirit to the construction of a free group.

In mathematics, a **cyclic order** is a way to arrange a set of objects in a circle. Unlike most structures in order theory, a cyclic order is not modeled as a binary relation, such as "*a* < *b*". One does not say that east is "more clockwise" than west. Instead, a cyclic order is defined as a ternary relation [*a*, *b*, *c*], meaning "after a, one reaches b before c". For example, [June, October, February]. A ternary relation is called a cyclic order if it is cyclic, asymmetric, transitive, and total. Dropping the "total" requirement results in a partial cyclic order.

The **Knuth – Bendix completion algorithm** is a semi-decision algorithm for transforming a set of equations into a confluent term rewriting system. When the algorithm succeeds, it effectively solves the word problem for the specified algebra.

In geometric topology, a field within mathematics, the obstruction to a homotopy equivalence of finite CW-complexes being a simple homotopy equivalence is its **Whitehead torsion** which is an element in the **Whitehead group** . These concepts are named after the mathematician J. H. C. Whitehead.

In mathematical group theory, the **Schur multiplier** or **Schur multiplicator** is the second homology group of a group *G*. It was introduced by Issai Schur (1904) in his work on projective representations.

In abstract algebra, the **conjugacy problem** for a group *G* with a given presentation is the decision problem of determining, given two words *x* and *y* in *G*, whether or not they represent conjugate elements of *G*. That is, the problem is to determine whether there exists an element *z* of *G* such that

In mathematics, one method of defining a group is by an **absolute presentation**.

In group theory, a **word** is any written product of group elements and their inverses. For example, if *x*, *y* and *z* are elements of a group *G*, then *xy*, *z*^{−1}*xzz* and *y*^{−1}*zxx*^{−1}*yz*^{−1} are words in the set {*x*, *y*, *z*}. Two different words may evaluate to the same value in *G*, or even in every group. Words play an important role in the theory of free groups and presentations, and are central objects of study in combinatorial group theory.

In mathematics, a group is called **boundedly generated** if it can be expressed as a finite product of cyclic subgroups. The property of bounded generation is also closely related with the congruence subgroup problem.

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 one of the main tools used in studying free groups,. They were introduced in to prove that every subgroup of a free group is free, but are now used in a variety of mathematics, including computational group theory, k-theory, and knot theory. The textbook devotes all of chapter 3 to Nielsen transformations.

In mathematical group theory, the **automorphism group of a free group** is a discrete group of automorphisms of a free group. The quotient by the inner automorphisms is the outer automorphism group of a free group, which is similar in some ways to the mapping class group of a surface.

In geometry, **Coxeter notation** is a system of classifying symmetry groups, describing the angles between fundamental reflections of a Coxeter group in a bracketed notation expressing the structure of a Coxeter-Dynkin diagram, with modifiers to indicate certain subgroups. The notation is named after H. S. M. Coxeter, and has been more comprehensively defined by Norman Johnson.

- Roger C. Lyndon, Paul E. Schupp,
*Combinatorial Group Theory*, Springer, 2001. ISBN 3-540-41158-5.

**Roger Conant Lyndon** 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 and the Lyndon–Hochschild–Serre spectral sequence.

The **International Standard Book Number** (**ISBN**) is a numeric commercial book identifier which is intended to be unique. Publishers purchase ISBNs from an affiliate of the International ISBN Agency.

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