# Grigorchuk group

From Groupprops

This article is about a particular group, i.e., a group unique upto isomorphism. View specific information (such as linear representation theory, subgroup structure) about this groupView a complete list of particular groups (this is a very huge list!)[SHOW MORE]

## Contents

## Definition

### As a subgroup of the automorphism group of the infinite rooted binary tree

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### As a group of Lebesgue measure-preserving transformations

## Arithmetic functions

Function | Value | Similar groups | Explanation |
---|---|---|---|

order | infinite | ||

exponent | infinite | although every element has finite order (and in fact all orders are powers of 2) there is no common bound on the orders of all elements. | |

minimum size of generating set | 3 | ||

underlying prime of p-group | 2 |

## Group properties

Property | Satisfied? | Explanation | Comment |
---|---|---|---|

abelian group | No | ||

nilpotent group | No | ||

solvable group | No | ||

2-generated group | No | ||

finitely generated group | Yes | follows from being 3-generated. | |

residually finite group | Yes | ||

Hopfian group | Yes | finitely generated and residually finite implies Hopfian | |

finitely presented group | No | Grigorchuk group is not finitely presented | |

group with solvable word problem | Yes | ||

group with solvable conjugacy problem | Yes | ||

LERF group (also called subgroup-separable group) | Yes | ||

p-group | Yes | Here, , i.e., the group is a 2-group | |

periodic group | Yes | Follows from being a 2-group, order of every element is a finite power of 2 |