# Word problems for groups

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*Gruppentheoret5ische Studien*Ⓣ contains the first appearance of a group presentation as we know it. He assumes (without rigorous justification) the existence of a free group of every finite rank and he gives a result (again not rigorously proved) that every m generator group can be obtained from the free group of rank m by adding relations. He knew intuitively how to tell whether a word in the free group is trivial - make all cancellations of

*aa*

^{-1}and

*a*

^{-1}

*a*for every generator

*a*appearing in the word.

Heinrich Tietze published the paper *Über die topologischen Invarianten mehrdimensionaler Mannigfaltigkeiten* Ⓣ in 1908. The topological part of this paper was based on the notion of the fundamental group introduced by Poincaré in 1895. In the 1908 paper Tietze defined the fundamental group of a manifold and showed that it was a topological invariant. To do this he introduced Tietze transformations. He proved that any two finite presentations for the same group can be transformed to each other by applying finitely many Tietze transformations. He wrote:-

In 1910 Max Dehn publishedOne observes at once that it may happen that two groups are isomorphic although they are defined using different systems of generators and defining relations. ... However, neither the general problem of characterising the totality of all abstract ways of generating a given group nor even the special problem of finding a method to decide whether two groups given by their presentations are isomorphic, have been solved.

*Über die Topologie des dreidimensionalen Raumes*Ⓣ. In this paper he considered the problem of when two knots are the same, building on the work of Poincaré. Here he dealt with the fundamental group as a key invariant and this group occurs naturally as given by a presentation. He quickly realised that the knot theory problems were special cases of much more general questions about finitely presented groups. He made the group theory problems explicit in his 1911 paper

*Über unendliche diskontinuierliche Gruppen*Ⓣ. We quote from the paper giving Dehn's formulation of the word problem for groups:-

Notice that there is a simple connection between the Conjugacy Problem and the Word Problem. If one can solve the Conjugacy Problem in a particular group then one can solve the Word Problem. For if a word w is conjugate to the identity, then it is equal to the identify of the group. Dehn also proved in this 1911 paper that a finitely presented group can have a subgroup which is not finitely presented. He also solved the isomorphism problem and the conjugacy problem for finitely presented groups with the property that each generator occurs at most twice in each of the defining relations.The general discontinuous group is given by n generators and m relations between them, as defined by Dyck(Math. Ann.,20and22). The results of those works, however, relate essentially to finite groups. The general theory of groups defined in this way at present appears very undeveloped in the infinite case. Here there are above all three fundamental problems whose solution is very difficult and which will not be possible without a penetrating study of the subject.

1. The Identity Problem[now called the Word Problem]: An element of the group is given as a product of generators. One is required to give a method whereby it may be decided in a finite number of steps whether this element is the identity or not.

2. The Transformation Problem[now called the Conjugacy Problem]: Any two elements S and T of the group are given. A method is sought for deciding the question of whether S and T can be transformed into each other, i.e. whether there is an element U of the group satisfying the relationS=UTU^{-1}.

3. The Isomorphism Problem: Given two groups, one is to decide whether they are isomorphic or not(and further, whether a given correspondence between the generators of one group and elements of the other is an isomorphism or not).

These problems have very different degrees of difficulty. ... One is already led to them by necessity with work in topology. Each knotted space curve, in order to be completely understood, demands the solution of the three above problems in a special case.

In 1912 Dehn studied the word problem and the conjugacy problem for the fundamental groups of orientable closed 2-dimensional manifolds. These are 1-relator groups, that is groups with a presentation with a single defining relation. He showed in *Transformationen der Kurven auf zweiseitigen Flächen* Ⓣ (1912) that in special cases one could solve the word problem using a direct approach - called Dehn's algorithm today. In this case one could construct a finite list of words in the group generators, *u*_{1} , *v*_{1} , *u*_{2} , *v*_{2} , ... *u*_{n} , *v*_{n} having the following properties. One has *u*_{i} = *v*_{i} as elements of the group, *v*_{i} is shorter than *u*_{i} for each *i*, and one has the property that if *w* is any word in the generators which represents the identity element, then at least one of the *u*_{i} is a subword of *w*. When a list with these properties exists one can easily solve the word problem. For, given a word *w* one looks to see if it contains a u_{i} . If it does not, then *w* is not the identity. If it does contain a *u*_{i} then replace *u*_{i} by *v*_{i}*t* o obtain a new word *w*'. Then continue the process, looking to see if *w*' contains a *u*_{i} . The process must terminate in a finite number of steps, since each time a *u*_{i} is replaced by a *v*_{i} a shorter word is obtained.

Dehn knew that the word problem was difficult and posed an entirely new type of question in mathematics. He wrote:-

Dehn solved the word problem for the trefoil knot group in 1914. He used his solution to show that right and left trefoils are distinct. We note, as an aside, that Waldhausen solved the word problem for all knot groups in 1968.Solving the word problem for all groups may be as impossible as solving all mathematical problems.

We now want to follow two routes forward, one for free groups and one for 1-relator groups. First note that the word problem in a free group is easily solved. As von Dyck noted, just use free reduction (but more is required for a rigorous proof which we will discuss below).

In 1921 Jakob Nielsen published *Om Regning med ikke kommutative Faktoren og dens Anvendelse in Gruppeteorien* Ⓣ. In this paper he showed how, given a set of words { *u*_{1} , *u*_{2} , ..., *u*_{m} } in a free group, one could construct a set of free generators for the subgroup they generate. In particular, he could deduce that a finitely generated subgroup of a free group was necessarily free. His work was motivated by the study of word problems. He also gave a method for solving the membership problem in free groups, that is an algorithm to determine whether a given word *w* in a free group was contained in the subgroup generated by { *u*_{1} , *u*_{2} , ..., *u*_{m} }.

A general statement of the membership problem was not given until 1958 by Mihailova:

*G*= <

*X*|

*R*> be a finite presentation, and

*U*a finite set of words {

*u*

_{1},

*u*

_{2}, ...,

*u*

_{m}} in the alphabet

*X*. Let

*H*be the subgroup of

*G*generated by

*U*. Given a word

*w*in the generators

*X*, can one decide whether or not

*w*is in

*H*?

*u*

_{1},

*u*

_{2}, ...,

*u*

_{m}} is a subset of

*X*.

Otto Schreier attended a lecture by Kurt Reidemeister in Hamburg in January 1926. In this lecture Reidemeister described how to find a presentation for a normal subgroup of finite index in a finitely presented group. After the seminar, Schreier thought about Reidemeister's method and eventually saw how to extend the method to arbitrary subgroups of possibly infinite index in a countably generated group. With this method he was able to extend Nielsen's result on subgroups of free groups to prove that every subgroup of a free group is free. He published these results in 1927 and at the same time gave a simple rigorous proof of the solution of the word problem in a free group. In this 1927 paper he also introduced the notion of free products with amalgamation.

Nielsen returned to the problem of a free group of countable rank in 1955. He noted that if *G* = < *X* | *R* > and *F* is the free group on the set *X*, then one can define a natural homomorphism from *F* to *G*. Suppose *K* is the kernel of this homomorphism, so *K* is the normal closure of the set of words *R* in the free group *F*. Then the word problem for *G* is equivalent to the problem of deciding which elements of *F* belong to *K*. He wrote:-

In 1926 Emil Artin solved the word problem for braid groups.If K is determined by a given infinite setS= {a_{1},a_{2}, .... }of elements generating K, then there exists a function L(n)such that all elements of K of length not exceeding n are contained in the subgroup K{_{L}generated by the subseta_{1},a_{2}, ....,a_{L}}of S; and forK_{L}a basis can be constructed in a finite number of steps. However, the dependence of L on n cannot, in general, be determined unless some further information can be gained from the structure of S. It is therefore only for finitely generated groups K that the identification of their elements can be assured unconditionally in a finite number of tests.

Let us now look at 1-relator groups. The solution to the word problem for these groups began with Dehn who stated the Freiheitssatz:

*G*= <

*A*|

*r*= 1 > where

*r*is freely and cyclically reduced. Let

*U*be a subset of

*A*with the property that not every generator appearing in

*r*is in

*U*. Then

*U*freely generates a free subgroup of

*G*.

We have given several examples of the word problem being proved for certain classes of groups, but said nothing about groups for which the word problem could not be solved. Until the 1950s only positive results could be obtained since totally new methods were needed even to state the problem of finding a group with insoluble word problem with the formal precision needed to give mathematicians a chance to come up with proofs. As Chandler and Magnus state:-

However, if one only 'knew one when one saw one', how could someone possible prove that no such algorithm existed. It required computability theory and developments in mathematical logic to even make the questions precise, but these areas were to not only provide explicit questions, they also provided solutions to the questions.The answer to a decision problem is an algorithm or a general and effective procedure. All one could say about these concepts in the time of Hilbert and Dehn was that one knew one when one saw one.

In the 1930s Kurt Gödel investigated how symbolic manipulation in formal logic could be simulated by functions on the natural numbers. These functions were built up from simple functions. Independently of Gödel, Alonzo Church was developing the *λ*-calculus designed to clarify the foundations of mathematics, in particular the meaning of variables. This development was to lead to the study of recursive functions. Stephen Kleene was a doctoral student of Church at Princeton and he was given as a project the task of determining which functions were l-definable. At first Kleene could make no progress at all on this project, then one day he had to visit the dentist to get two wisdom teeth extracted. While sitting in the dentist's chair waiting for this unpleasant experience, inspiration struck and suddenly he saw the route to the solution. Within six months he had complete answers to the problem. In 1933 Church produced what today is called "Church's thesis" which states that l-definability is the precise notion of "computable function". This is not a theorem, it is simply saying that the vague notion of "computable function" can be given the rigorous definition of l-definability. In 1936 Alan Turing produced the notion of a Turing machine and proposed that a "computable function" was one which could be computed by a Turing machine. Quite quickly it was shown that a function was l-definable precisely when it can be computed by a Turing machine, so the two concepts are equivalent. This was enough for almost all mathematicians to accept that these equivalent concepts rigorously embodied the intuitive concept of "computable function".

*P*where

*P*is an infinite set of questions

*Q*

_{i}, we say that

*P*is solved if we can compute the function

*f*:

*P*→ {

*Y*,

*N*} so that

*f*(

*Q*

_{i}) =

*Y*if the answer to

*Q*

_{i}is YES, and

*f*(

*Q*

_{i}) =

*N*if the answer to

*Q*

_{i}is NO. If we can show that no such function

*f*can be computed by a Turing machine, then problem

*P*is insoluble. Before continuing to describe the history, let us define some concepts.

A set *S* is recursively enumerable if there is an algorithm to list the elements of *S*, where we use the standard Turing machine model of computation.

A set *S* is recursive if both *S* and its complement are recursively enumerable.

In 1938 Church proposed that the word problem for groups should be proved insoluble using the new rigorous definitions of computable. In 1946 and 1947 the first breakthrough was made by Emil Post. Now Post had been unbelievably unlucky. He had produced most of the ideas of Gödel and Church before they did, but his work had not been considered publishable. It is often the case that when someone comes up with a really revolutionary idea, the world is not ready for it and this seems to have been the case here. Post continued, despite the blow of failing to get his work published, and came up with the idea of a Turing machine at about the same time as Turing. However Turing's work was quickly published before Post who, given his bad experience with publishing, was not rushing into print. When Post did publish, five years after Turing, he gave what is today called the Post Correspondence Problem.

*A*= {

*a*

_{1},

*a*

_{2}, ....,

*a*

_{n}} and an infinite set

*S*of pairs of words (

*u*

_{i},

*v*

_{i}), where

*u*

_{i},

*v*

_{i}are words in

*A*. Determine if any pair in

*S*has the two components equal.

If Post's semigroup had been embeddable in a group then the word problem for groups would have been solved in the negative at the same time. However, this was not the case and the word problem for groups remained open. Turing learnt about the word problem for groups after reading Post's paper. He thought about the problem for ten days, then declared he had solved it. He arranged to give a seminar describing the proof, but just before the seminar he found an error. He was left with something, however, he had proved that the word problem is insoluble for cancellative semigroups.

In 1954 Bill Boone published a paper in which he defined the quasi-Magnus problem and proved it insoluble.

*G*be given by a finite presentation <

*A*|

*R*> . Decide whether a word

*w*in the alphabet

*A*is equal, as an element of

*G*, to a word

*u*in

*A*whose exponents are all positive (i.e. no inverses of generators appear in the word

*u*).

However, a different approach was to prove highly significant. Graham Higman, Bernhard Neumann and Hanna Neumann introduced what is now called an HNN extension in 1949. Higman made a major breakthrough using HNN extensions. First a definition:

*G*is recursively presented if it has a presentation <

*A*|

*R*> where

*R*is a recursively enumerable set of words.

*H*can be embedded in a finitely presented group if and only if

*H*is recursively presented.

*H*(

*S*) = <

*a*,

*b*,

*c*,

*d*|

*a*

^{-i}

*ba*

^{i}=

*c*

^{-i}

*dc*

^{i},

*i*∈

*S*> where

*S*is a recursively enumerable set which is not recursive. Then

*H*(

*S*) is the free product of the free group <

*a*,

*b*> with the free group <

*c*,

*d*> with the subgroups <

*a*

^{-i}

*ba*

^{i}|

*i*

*in*

*S*> and <

*c*

^{-i}

*dc*

^{i}|

*i*∈

*S*> amalgamated. Then

*a*

^{-i}

*ba*

^{i}

*c*

^{-i}

*d*

^{-1}

*c*

^{i}= 1 in

*H*(

*S*) if and only if

*i*is in

*S*. But this is not decidable. Hence

*H*(

*S*) has insoluble word problem. Finally use Higman embedding to embed

*H*(

*S*) in a finitely presented groups

*G*(

*S*) to obtain a finitely presented group with an insoluble word problem.

We note that Higman also used his embedding theorem to prove that there exists a finitely presented group *U* such that given any finitely presented group *G* then *G* is isomorphic to subgroup of *U*.

Sergei Ivanovich Adian, a student of Petr Sergeevich Novikov, proved a whole range of decision problems in 1957 and 1958 based on the idea of a Markov property.

*P*of finitely presented groups is a Markov property if there are two finitely presented groups

*G*(+) and

*G*(-) such that

(a) *G*(+) has property *P*

(b) if *G*(-) is embedded in a finitely presented group *G*, then *G* does not have property *P*.

*G*(+) to be the cyclic group of order 2 and

*G*(-) to be the infinite cyclic group.

*P*of finitely presented groups is recursively recognisable if, given an arbitrary finite presentation, it is computable (with a Turing machine) whether or not the group with this presentation has the property

*P*. In other words there is an algorithm to determine whether the group with a given presentation satisfies property

*P*.

**Theorem**:

If

*P*is a Markov property of finitely presented groups, then

*P*is not recursively recognisable.

**Corollary**:

The following properties of finitely presented groups are not recursively recognisable, namely being

(ii) finite;

(iii) abelian;

(iv) nilpotent;

(v) soluble;

(vi) free;

(vii) torsion free;

(viii) residually finite;

(ix) having soluble word problem;

(x) simple.

One also obtains as a corollary to Adian's theorem that the isomorphism problem is insoluble, since one cannot determine whether a group is isomorphic to the trivial group.

In 1959 Gilbert Baumslag, Bill Boone and Bernhard Neumann proved the following theorem:

**Theorem**:

There is a finite presented group

*G*such that there is no algorithm to determine whether or not a word in the given generators represents

*G*;

(ii) an element which commutes with a given element;

(iii) an

*n*th power, where

*n*> 1 is fixed;

(iv) an element with finitely many conjugates;

(v) a commutator;

(vi) an element of finite order.

**Theorem**:

Let

*F*be a free group of rank 2 and let

*G*be the direct product of

*F*with itself. Then

*G*has a subgroup

*L*for which the membership problem is insoluble.

*G*=

*F*×

*F*further proving that one cannot decide whether

*L*=

*G*. Also he proved that

*G*has a subgroup

*L*with insoluble conjugacy problem. This gives groups with soluble word problem but insoluble conjugacy problem.

**Article by:** *J J O'Connor* and *E F Robertson*

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