This term is related to: combinatorial group theory
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This term is related to: geometric group theory
View other terms related to geometric group theory  View facts related to geometric group theory
Definition
A group with solvable word problem is a finitely generated group satisfying the following equivalent conditions:
 There is a finite generating set and an algorithm that, given any word in terms of the generators, can determine in finite time whether or not that word equals the identity element of .
 For any finite generating set there is an algorithm that, given any word in terms of the generators, can determine in finite time whether or not that word equals the identity element of .
 For any algebraically closed group , is isomorphic to some subgroup of .
Note that the finite time that the algorithm takes to terminate depends on the word itself. However, since the generating set is finite, there are only finitely many words of a given length, and we can hence obtain a bound on the time the algorithm takes, that depends only on the length of the word. However, that bound may not in general be a computable function of the length.
This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
View a complete list of group properties
VIEW RELATED: Group property implications  Group property nonimplications Group metaproperty satisfactions  Group metaproperty dissatisfactions  Group property satisfactions  Group property dissatisfactions
Relation with other properties
Stronger properties
Property 
Meaning 
Proof of implication 
Proof of strictness (reverse implication failure) 
Intermediate notions

finitely generated free group 



Automatic group, Finitely presented conjugacyseparable group, Finitely presented residually finite group, Group satisfying a quadratic isoperimetric inequality, Group with a finite complete rewriting system, Group with nondeterministic polynomialtime solvable word problem, Group with polynomialtime solvable word problem, Group with solvable conjugacy problem, Wordhyperbolic groupFULL LIST, MORE INFO

onerelator group 
finitely presented with only one relation 
onerelator implies solvable word problem 

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finitely generated abelian group 



Finitely presented conjugacyseparable group, Finitely presented residually finite group, Group satisfying a quadratic isoperimetric inequality, Group with a finite complete rewriting system, Group with nondeterministic polynomialtime solvable word problem, Group with polynomialtime solvable word problem, Group with solvable conjugacy problemFULL LIST, MORE INFO

finitely generated nilpotent group 



Group with a finite complete rewriting system, Group with polynomialtime solvable word problemFULL LIST, MORE INFO

polycyclic group 



Group with polynomialtime solvable word problemFULL LIST, MORE INFO

finitely presented residually finite group 

finitely presented and residually finite implies solvable word problem 

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finitely presented conjugacyseparable group 

(via finitely presented residually finite) 

Finitely presented residually finite group, Group with solvable conjugacy problemFULL LIST, MORE INFO

wordhyperbolic group 



Automatic group, Group with polynomialtime solvable word problemFULL LIST, MORE INFO

automatic group 



Template:Intermediate notion short

biautomatic group 



Automatic group, Group satisfying a polynomial isoperimetric inequality, Group satisfying a quadratic isoperimetric inequality, Group with nondeterministic polynomialtime solvable word problem, Group with polynomialtime solvable word problemFULL LIST, MORE INFO

group with solvable conjugacy problem 



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group with a finite complete rewriting system 



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combable group 



Group satisfying a polynomial isoperimetric inequality, Group satisfying a quadratic isoperimetric inequality, Group with nondeterministic polynomialtime solvable word problemFULL LIST, MORE INFO

group with polynomialtime solvable word problem 



Group with nondeterministic polynomialtime solvable word problemFULL LIST, MORE INFO

Weaker properties
Property 
Meaning 
Proof of implication 
Proof of strictness (reverse implication failure) 
Intermediate notions

finitely generated group 



