On the unsolvability of the conjugacy problem for subgroups of the group \(R_5\) of pure braids (Q1966204)

The braid group \(B_{n+1}\) is defined by the generators \(\sigma_1,\dots,\sigma_n\) and the defining relations \[ \sigma_i\sigma_j\sigma_i=\sigma_j\sigma_i\sigma_j\;(|i-j|=1),\quad\sigma_i\sigma_j=\sigma_j\sigma_i\;(|i-j|>1). \] There exists a homomorphism of the group \(B_{n+1}\) to the permutation group \(S_{n+1}\) that maps the generators \(\sigma_i\) to the transpositions \((i,i+1)\). A braid that realizes the identity permutation is said to be pure. The subgroup of pure braids of the group \(B_{n+1}\) is denoted by \(R_{n+1}\). \textit{I. V. Dobrynina} [Algorithmic problems of the theory of groups and semigroups, Tul'sk. Gos. Ped. Inst., Tula 1994, 62-70 (1994)] proved the unsolvability of the conjugacy problem for subgroups of the groups \(R_{n+1}\) for \(n>4\). In this paper the authors prove the same result for \(n=4\).