Braid group

Remarks

Geometrically, $\sigma_i$ is the braid in which strand $i$ crosses over strand $i + 1$ and the other strands go straight down. By a theorem of Artin, this presentation gives the group of isotopy classes of geometric braids on $n$ strands. It is the Artin group of the Coxeter matrix of $S_n$.

Example: $B_1$ is trivial and $B_2 = \langle \sigma_1 \mid \ \rangle \cong \Z$. The group $B_3 = \langle \sigma_1, \sigma_2 \mid \sigma_1 \sigma_2 \sigma_1 = \sigma_2 \sigma_1 \sigma_2 \rangle$ is not abelian. The map $\sigma_i \mapsto (i \ i{+}1)$ respects both relations, so it gives a surjective homomorphism $B_n \to S_n$ onto the symmetric group, and $S_3$ is not abelian. The kernel of $B_n \to S_n$ is the pure braid group.

Non-example: the symmetric group $S_n$ with the transpositions $(i \ i{+}1)$ satisfies relations 1 and 2, but it is not $B_n$ for $n \ge 2$. In $S_n$ the generators also satisfy $(i \ i{+}1)^2 = 1$, while in $B_n$ each $\sigma_i$ has infinite order, since the map $B_n \to \Z$ with $\sigma_i \mapsto 1$ respects the relations and sends $\sigma_i^k$ to $k$.