Grigorchuk group

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VIEW DEFINITIONS USING THIS AS EXAMPLE: All terms |Group properties |Subgroup properties|
VIEW FACTS USING THIS AS EXAMPLE: All facts |Group property non-implications |Subgroup property non-implications |Group metaproperty dissatisfactionsSubgroup metaproperty dissatisfactions

Definition

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

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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, $p = 2$, 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