Marcelo and Tom write We show that Thompson's group F is the symmetry group of the "generic idempotent". That is, take the monoidal category freely generated by an object A and an isomorphism A \otimes A --> A; then F is the group of automorphisms of A. Tom has pointed out to me that the review of the old Freyd/Heller I posted give no hint of its relevance. Therefor this: F was defined (40 years ago) as the initial model for a group with an endomorphism that's conjugate to its square. More formally: consider the equational theory that adds to the theory of groups a constant, s, and a unary operator e, subject to two further equations: e(xy) = (ex)(ey) "e is a endomorphism" s(ex) = (e(ex))s "e is a conjugacy-idempotent" The initial algebra for this theory is the group F. (If one insists on removing the type-error in the last sentence, then try "the initial algebra for this theory when subjected to the forgetful functor back to groups is F.") If one defines a sequence of elements s_n = e^n(s) they clearly generate F (as a group) and it isn't hard to see that a complete set of relations for F (as a group) is the doubly-infinite family s_a s_b = s_{b+1} s_a one such equation for each a < b. (It took me ten years to find a proof that just two of these equations imply all the others.)