# Braid Groups

## `BraidGroup(W): GrpFPCox -> GrpFP, Map`

The braid group $B$ of the Coxeter group $W$ as a finitely presented group, together with the natural map $W\to B$. Words in the braid group are not automatically normalised. However, the braid group of type $A_n$ with normalisation can be constructed with the command `BraidGroup(n+1)` (see Chapter [Braid Groups](../../FinitelyPresentedGroups/BraidGroups/index-braid-groups.md#chapgrpbrd)).

## `PureBraidGroup(W): GrpFPCox -> GrpFP, Map`

Returns the pure braid group of the Coxeter group $W$, ie. the kernel of the epimorphism from the braid group of $W$ to $W$. Words in the pure braid group are not automatically normalised.

## `Example: Braid Groups (ex-a9e54b)`

```magma
> W<a,b,c> := CoxeterGroup(GrpFPCox, "B3");
> W;
Coxeter group: Finitely presented group on 3 generators
Relations
    a * b * a = b * a * b
    a * c = c * a
    (b * c)^2 = (c * b)^2
    a^2 = Id($)
    b^2 = Id($)
    c^2 = Id($)
> B<x,y,z> := BraidGroup(W);
> B;
Finitely presented group B on 3 generators
Relations
    x * y * x = y * x * y
    x * z = z * x
    (y * z)^2 = (z * y)^2
> P := PureBraidGroup(W);
> P;
Finitely presented group P on 3 generators
Generators as words in group B
    P.1 = x^2
    P.2 = y^2
    P.3 = z^2

```