TITLE: An H-group that's not a loop space?
QUESTION [10 upvotes]: 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.

REPLY [10 votes]: The first example appears in the paper "Homotopy associativity and finite
CW complexes" Topology vol.9 (1970) 121-128 by Alexander Zabrodsky. With
present knowledge it is possible to construct many examples.
Here are some examples.  Consider Sp(n) with n 3 or more, or G_2.  Construct X from a pullback diagram of p-local spaces, where X localized at 2 or 3 is the Lie group and localized at the remaining primes is the product of local sphere having the same type as the Lie group. Then X is homotopy associative because each of its localizations is homotopy associative. The space X is not a mod p group for p between 5 and 2n  if Sp(n) is used and not a mod 5 group if G_2 is used in the construction. The latter claim is based of a theorem proved by Clarence Wilkerson, "K-theory Operations in mod-p Loop Spaces"  Math.Z. 132, 29-44, (1973)