Some subgroups of Artin's braid group (Q1361372)

The braid group \(B_n\) has the presentation \[ \langle\sigma_1,\dots,\sigma_{n-1};\;\sigma_i=\sigma_j\;(|i-j|>1),\;\sigma_i\sigma_j\sigma_i=\sigma_j\sigma_i\sigma_j\;(|i-j|=1)\rangle. \] Let \(S_n\) denote the group of permutations of the set \(A_n=\{1,\dots,n\}\). There exists the standard homomorphism \(h\colon B_n\to S_n\), such that \(h(\sigma_i)=(i,i+1)\). Let \(P\) denote a partition of the set \(A_n\): \(P=\{A^1_n,\dots,A^k_n\}\), where \(\bigcup A_n^i=A_n\), \(A_n^i\cap A_n^j=\emptyset\). Let \(B_{n,P}\) denote the subgroup consisting of all braids \(b\) such that \(h(b)\) preserves the partition \(P\). Using the Reidemeister-Schreier method, the author calculates presentations for the subgroups \(B_{n,P}\).