# Braid group:B3

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]

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

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 |