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We study the permutation group $\mathcal{BK}_P$ generated by Bender-Knuthmoves on linear extensions of a poset $P$, an analog of the Berenstein-Kirillovgroup on column-strict tableaux. We explore the group relations, with anemphasis on identifying posets $P$ for which the cactus relations hold in$\mathcal{BK}_P$. We also examine $\mathcal{BK}_P$ as a subgroup of thesymmetric group $\mathfrak{S}_{\mathcal{L}(P)}$ on the set of linear extensionsof $P$ with the focus on analyzing posets $P$ for which $\mathcal{BK}_P =\mathfrak{S}_{\mathcal{L}(P)}$.

Bender-Knuth involutions on linear extensions of posets