The symmetric group, in terms of presentation, is given by a group with generators $x_1,x_2,\cdots,x_n$ with following types of relations:
(R1) $x_1x_{i+1}x_i=x_{i+1}x_ix_{i+1}$
(R2) $x_ix_j=x_jx_i$ for $|i-j|\geq 2$
(R3) $x_i^2=1$
There is a class of groups which are closely related with this group. The Braid group $B_n$ has presentation:
$$B_n=\langle x_1,x_2,\cdots,x_n : R1, R2\rangle.$$ In other words, dropping relations (R3) from above presentation of symmetric group, we get Braid group. Then my question is
Question: If we drop relations (R1) or (R2) from presentation of $S_n$, what kind of groups we get? Are they finite? Are these groups studied well?