Editing Word problem for groups (section)

== History ==

Throughout the history of the subject, computations in groups have been carried out using various [[Normal form (abstract rewriting)|normal forms]]. These usually implicitly solve the word problem for the groups in question. In 1911 [[Max Dehn]] proposed that the word problem was an important area of study in its own right,{{sfn|Dehn|1911}} together with the [[conjugacy problem]] and the [[group isomorphism problem]]. In 1912 he gave an algorithm that solves both the word and conjugacy problem for the [[fundamental group]]s of closed orientable two-dimensional manifolds of genus greater than or equal to 2.{{sfn|Dehn|1912}}  Subsequent authors have greatly extended [[Small cancellation theory#Dehn's algorithm|Dehn's algorithm]] and applied it to a wide range of group theoretic [[decision problem]]s.<ref>{{Citation|last=Greendlinger|first=Martin|date=June 1959|title=Dehn's algorithm for the word problem|journal=Communications on Pure and Applied Mathematics|volume=13|issue=1|pages=67–83|doi=10.1002/cpa.3160130108|postscript=.}}</ref><ref>{{Citation|last=Lyndon|first=Roger C.|author-link=Roger Lyndon|date=September 1966|title=On Dehn's algorithm|journal=Mathematische Annalen|volume=166|issue=3|pages=208–228|doi=10.1007/BF01361168|hdl=2027.42/46211|s2cid=36469569|postscript=.|url=http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=GDZPPN002296799&L=1|hdl-access=free}}</ref><ref>{{Citation|author-link1=Paul Schupp|last1=Schupp|first1=Paul E.|date=June 1968|title=On Dehn's algorithm and the conjugacy problem|journal=Mathematische Annalen|volume=178|issue=2|pages=119–130|doi=10.1007/BF01350654|s2cid=120429853|postscript=.|url=http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=GDZPPN002300036&L=1}}</ref>

It was shown by [[Pyotr Novikov]] in 1955 that there exists a finitely presented group <math>G</math> such that the word problem for <math>G</math> is [[Undecidable problem|undecidable]].<ref>{{Citation|last=Novikov|first=P. S.|author-link=Pyotr Novikov|year=1955|title=On the algorithmic unsolvability of the word problem in group theory|language=ru| zbl=0068.01301 | journal=[[Proceedings of the Steklov Institute of Mathematics]]|volume=44|pages=1–143}}</ref> It follows immediately that the uniform word problem is also undecidable. A different proof was obtained by [[William Boone (mathematician)|William Boone]] in 1958.<ref>{{Citation|last=Boone|first=William W.| author-link=William Boone (mathematician) | year=1958|title=The word problem|journal=Proceedings of the National Academy of Sciences|volume=44|issue=10|pages=1061–1065|url=http://www.pnas.org/cgi/reprint/44/10/1061.pdf|doi=10.1073/pnas.44.10.1061|zbl=0086.24701 |pmc=528693|pmid=16590307|bibcode=1958PNAS...44.1061B|doi-access=free}}</ref>

The word problem 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, [[abstract algebra|algebra]]. As a result of its unsolvability, several other problems in combinatorial group theory have been shown to be unsolvable as well.

The word problem is in fact solvable for many groups <math>G</math>. For example, [[polycyclic group]]s have solvable word problems since the normal form of an arbitrary word in a polycyclic presentation is readily computable; other algorithms for groups may, in suitable circumstances, also solve the word problem, see the [[Todd–Coxeter algorithm]]<ref>{{citation |last1=Todd |first1=J. |last2=Coxeter |first2=H.S.M. |title=A practical method for enumerating cosets of a finite abstract group |journal=Proceedings of the Edinburgh Mathematical Society |volume=5 |issue=1 |pages=26–34 |date=1936 |doi=10.1017/S0013091500008221 |url=|doi-access=free }}</ref> and the [[Knuth–Bendix completion algorithm]].<ref>{{citation |first1=D. |last1=Knuth |first2=P. |last2=Bendix |chapter=Simple word problems in universal algebras |chapter-url=https://books.google.com/books?id=KIDiBQAAQBAJ&pg=PA263 |editor-first=J. |editor-last=Leech |title=Computational Problems in Abstract Algebra: Proceedings of a Conference Held at Oxford Under the Auspices of the Science Research Council Atlas Computer Laboratory, 29th August to 2nd September 1967 |publisher=Springer |date=2014 |orig-year=1970 |isbn=9781483159423 |pages=263–297 |url=}}
</ref> On the other hand, the fact that a particular algorithm does not solve the word problem for a particular group does not show that the group has an unsolvable word problem. For instance Dehn's algorithm does not solve the word problem for the fundamental group of the [[torus]]. However this group is the direct product of two infinite cyclic groups and so has a solvable word problem.