Grigorchuk group
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Contents
Definition
As a subgroup of the automorphism group of the infinite rooted binary tree
PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]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, ![]() |
|
periodic group | Yes | Follows from being a 2-group, order of every element is a finite power of 2 |