Group with solvable word problem

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This term is related to: combinatorial group theory
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This term is related to: geometric group theory
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Definition

A group with solvable word problem is a finitely generated group G satisfying the following equivalent conditions:

  1. 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 G.
  2. 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 G.
  3. For any algebraically closed group K, G is isomorphic to some subgroup of K.

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 non-implications |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 conjugacy-separable group, Finitely presented residually finite group, Group satisfying a quadratic isoperimetric inequality, Group with a finite complete rewriting system, Group with nondeterministic polynomial-time solvable word problem, Group with polynomial-time solvable word problem, Group with solvable conjugacy problem, Word-hyperbolic group|FULL LIST, MORE INFO
one-relator group finitely presented with only one relation one-relator implies solvable word problem |FULL LIST, MORE INFO
finitely generated abelian group Finitely presented conjugacy-separable group, Finitely presented residually finite group, Group satisfying a quadratic isoperimetric inequality, Group with a finite complete rewriting system, Group with nondeterministic polynomial-time solvable word problem, Group with polynomial-time solvable word problem, Group with solvable conjugacy problem|FULL LIST, MORE INFO
finitely generated nilpotent group Group with a finite complete rewriting system, Group with polynomial-time solvable word problem|FULL LIST, MORE INFO
polycyclic group Group with polynomial-time solvable word problem|FULL LIST, MORE INFO
finitely presented residually finite group finitely presented and residually finite implies solvable word problem |FULL LIST, MORE INFO
finitely presented conjugacy-separable group (via finitely presented residually finite) Finitely presented residually finite group, Group with solvable conjugacy problem|FULL LIST, MORE INFO
word-hyperbolic group Automatic group, Group with polynomial-time solvable word problem|FULL 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 polynomial-time solvable word problem, Group with polynomial-time solvable word problem|FULL LIST, MORE INFO
group with solvable conjugacy problem |FULL LIST, MORE INFO
group with a finite complete rewriting system |FULL LIST, MORE INFO
combable group Group satisfying a polynomial isoperimetric inequality, Group satisfying a quadratic isoperimetric inequality, Group with nondeterministic polynomial-time solvable word problem|FULL LIST, MORE INFO
group with polynomial-time solvable word problem Group with nondeterministic polynomial-time solvable word problem|FULL LIST, MORE INFO

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
finitely generated group