Word problem for groups

In abstract algebra, the word problem for groups is the problem of deciding whether two given words of a presentation of a group represent the same element. There exists no general algorithm for this problem, as was shown by Pyotr Sergeyevich Novikov. The proof was announced in 1952 and published in 1955. A much simpler proof was obtained by Boone in 1959.

The word problem is only concerned with finitely presented groups, i.e. those groups which can be specified by finitely many generators and finitely many relations among those generators. A word is a product of generators, and two such words may denote the same element of the group even if they appear to be different, because by using the group axioms and the given relations it may be possible to transform one word into the other. The problem then is to find an algorithm which for any two given words decides whether they denote the same group element.

In more concrete terms, the problem can be expressed as a rewriting question, for literal strings. For a presentation P of a group G, P will specify a certain number of generators

x, y, z, ...

for G. We need to introduce one letter for x and another (for convenience) for the group element represented by x−1. Call these letters (twice as many as the generators) the alphabet A for our problem. Then each element in G is represented in some way by a product

abc ... pqr

of symbols from A, of some length, multiplied in G. The effect of the relations in G is to make various such strings represent the same element of G. In fact the relations provide a list of strings that can be either introduced where we want, or cancelled out whenever we see them, without changing the 'value', i.e. the group element that is the result of the multiplication.

For a simple example, take the presentation <x|x3>. Writing y for the inverse of x, we have possible strings of x′s and y′s. Whenever we see xxx, or xy or yx we may strike these out. We should also remember to strike out yyy; this says that since the cube of x is the identity element of G, so is the cube of the inverse of x. Under these conditions the word problem becomes easy. First reduce strings to e, x, xx, y or yy. Then note that we may also multiply by xxx, so we can convert yy to x. The result is that we can prove that the word problem here, for what is the cyclic group of order three, is soluble.

This is not, however, the typical case. For the example, we have a canonical form available that reduces any string to one of length at most three, by decreasing the length monotonically. In general, it is not true that one can get a canonical form for the elements, by stepwise cancellation. One may have to use relations to expand a string many-fold, in order eventually to find a cancellation that brings the length right down.

The upshot is, in the worst case, that the relation between strings that says they are equal in G is not decidable.

It is important to realize that the word problem is in fact solvable in many special cases; algorithms for many group presentations can be readily given. For example, see Todd-Coxeter algorithm and Knuth-Bendix completion algorithm. Novikov's result says that there are some finitely presented groups for which no algorithm solving the word problem exists.

The word problem is sometimes called the Dehn problem, after Max Dehn who first posed it in 1911. It was one of the first examples of an unsolvable problem to be found not in mathematical logic or the theory of algorithms, but in one of the central branches of classical mathematics, algebra. As a result of its unsolvability, several other problems in combinatorial group theory have been shown to be unsolvable as well.

References:

• Boone, Cannonito, Lyndon. Word Problems: Decision Problem in Group Theory. Netherlands: North-Holland. 1973.
• P. S. Novikov. On the algorithmic unsolvability of the word problem in group theory. Trudy Mat. Inst. Steklov 44 (1955), pp 1-143. (in Russian)
• W. W. Boone. The word problem. Annals of Mathematics, 70(2) (1959), pp 207-265
• J.J. Rotman. The Theory of Groups: An Introduction. Boston: Allyn and Bacon. 1965.
• J. Stillwell. The word problem and the isomorphism problem for groups. Bulletin AMS 6 (1982), pp 33-56