The whole point is that they didn't immediately set all letters equal to one. The only relations they added (on top of cancellation) was that identically soundi

The whole point is that they didn't immediately set all letters equal to one. The only relations they added (on top of cancellation) was that identically sounding words are equal within the monoid. From this, they were able to prove that the group must be the trivial group, no matter how you try to come up with a multiplication table.