Classifying spaces of $E_1$-spaces

Question

I'm a newbie trying to understand May's recognition principle on $E_1$-spaces. In May's paper The Geometry of Iterated Loop Spaces, the classifying space of an $E_1$-space $X$ is defined to be $B(\Sigma, E_1, X)$. Is there a simple way to see how this coincides with the usual classifying space $BG$ in the case where $X=G$ is a topological group?