# Grigorchuk group

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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