Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 4 / 3 / B

Here transitivity is order n-transitivity: increasing tuples are compared, rather than arbitrary reorderings. For increasing tuples ${\alpha }_1<\cdots <{\alpha }_n$ and ${\beta }_1<\cdots <{\beta }_n$, choose $h_{ij}$ sending both ${\alpha }_i$ to ${\beta }_i$ and ${\alpha }_j$ to ${\beta }_j$ when $i\ne j$, using order two-transitivity. For $i=j$, choose an element sending the single point to its target, using transitivity. Define

This belongs to the lattice-ordered permutation group. At ${\alpha }_i$, all terms in its $i$th inner meet equal ${\beta }_i$. At ${\alpha }_k$, every inner meet is at most ${\beta }_k$, because its $j=k$ term takes that value. The outer join is consequently exactly ${\beta }_k$ there. Thus $h$ maps the whole tuple to its target. This finite interpolation by lattice operations proves every finite order-transitivity without assuming the group permits arbitrary cut-and-paste restrictions of its elements.