@TimCampion and @MaximeRamzi, the "H-group" definition I found online that makes the maps I said work is that X be a "group up to homotopy" in the same sense that an H-space is a "unital magma up to homotopy." That is, one asks for a homotopy between the maps on X^3 with values x.(y.z) and (x.y).z and a weak "inverse" operation on X such that the functions on X taking x to i(x).x and x.i(x) are both homotopic to the identity. If X is a loop-space this is automatic from the lemmas in the standard introduction to the fundamental group, unless I'm missing something, but I assume all these noti…