Braid group:B3

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VIEW FACTS ABOUT THIS GROUP: All factsGroup property satisfactionsSubgroup property satisfactionsGroup property dissatisfactionsSubgroup property satisfactions
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

This group is defined as the braid group of degree three, i.e., the group $B_{3}$. Explicitly, it is given by the following presentation:

$\langle s_{1},s_{2}\mid s_{1}s_{2}s_{1}=s_{2}s_{1}s_{2}\rangle$

Up to isomorphism, it is also equivalent to the following:

• The knot group of the trefoil knot.
• The universal central extension of special linear group:SL(2,Z).

Group properties

Most of the properties below can be explained by the fact that the group admits free group:F2 as a subquotient.

Property	Satisfied?	Explanation	Comment
cyclic group	No
abelian group	No
nilpotent group	No
solvable group	No
simple group	No
finitely generated group	Yes
2-generated group	Yes
finitely presented group	Yes
Noetherian group	No