The description of the Coleman–Mandula theorem on Wikipedia starts with the following assumptions:
Every quantum field theory satisfying the assumptions,
Below any mass M, there is only a finite number of particle types.
Any two-particle state undergoes some reaction at almost all energies
The amplitude for elastic two-body scattering are analytic functions of scattering angle at almost all energies,
Then one has non-trivial interactions if the theory has a Lie group symmetry which is always a direct product of the Poincaré group and an internal group if there is a mass gap: no mixing between these two is possible. As the authors say in the introduction to the 1967 publication, "We prove a new theorem on the impossibility of combining space-time and internal symmetries in any but a trivial way."
Why are assumptions 2. and 3. taken for granted? Why are they necessary or sufficient?
Why is the presence of a mass gap an issue to establish a direct product of the Poincaré group and an internal group? If it is massless, can we establish instead a direct product of the conformal group and an internal group?