# Group with solvable conjugacy 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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## Contents

## Definition

### Symbol-free definition

A **group with solvable conjugacy problem** is a finitely presented group with a finite presentation having the following property: there is an algorithm that, given any two words in the generators, can, in finite time, test whether the two words represent conjugate elements in the group.

The finite time taken depends on the word, but because the generating set is finite, there are only finitely many words of any length, so we cna obtain an upper bound on the time taken by the algorithm as a function of the length of the word.

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

finitely generated free group | Finitely presented conjugacy-separable group|FULL LIST, MORE INFO | |||

finitely generated abelian group | Finitely presented conjugacy-separable group|FULL LIST, MORE INFO | |||

finite group | Finitely presented conjugacy-separable group|FULL LIST, MORE INFO | |||

finitely presented conjugacy-separable group | finitely presented and conjugacy-separable implies solvable conjugacy problem | |FULL LIST, MORE INFO |

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

group with solvable word problem | ||||

finitely presented group |