In other words, is the natural map $M \to \Omega B M$, for $M=JX$ the James construction on a space, a group completion? (By "group completion" I mean at the level of homology, I am aware of the space level version.) The versions of the group completion theorem that I have found such as Segal/McDuff have conditions on the monoid $M$, involving some kind of commutativity condition, and it is not clear to me that $JX$ satisfies any of these conditions. Could someone provide a reference or statement of a group completion theorem that clearly applies to $JX$, or is there actually a counterexample?
