Recall that an H-group is a space $X$ (in the sense of homotopy theory, so say CW complex) that is a group object in the homotopy category. I.e., there's a multiplication map $X \times X \rightarrow X$ which is associative up to homotopy and with an inverse map again up to homotopy. The standard example of an H-group is a loop space $\Omega X$. What is a simple example of an H-group that is not a loop space?

If I understand the language right, I'm asking for a group-like $A_3$ algebra that's not $A_\infty$. If I were asking for $A_2$ but not $A_3$ then I know $S^7$ is a good example.

My motivation is illustrating some of the subtleties in defining $\infty$-groups for a HoTT seminar.

Exercisesthere are four homotopy classes of map $RP^2\to RP^2$, and there are four homotopy classes of map $\Sigma RP^2 \to \Sigma RP^2$; but the suspension $RP^2\to \Omega\Sigma RP^2$ misses the degree-2 map. (otherwise the cofiber of something would have a nontrivial $Sq^2$ acting on $H^1$! ) Consequently, some of the four maps below get confused in the suspension, too. Upshot is that no map $\Omega\Sigma RP^2 \to RP^2 $ can be a retraction. $\endgroup$ – Jesse C. McKeown Aug 31 '16 at 2:39