I am looking for an elementary reference for the following fact.
Let $M$ be a topological monoid and suppose moreover that it is group-like, ie. $\pi_{0}(M)$ is a group. Then the canonical map $M \rightarrow \Omega BM$ is a weak equivalence.
I am also interested in the simplicial analogue of the above. I know this follows from group-completion, for example in the way it is stated in Goerss-Jardine, but this is supposed to be the trivial case and so I am looking for a simple argument.