Let A be a CW-complex and (X,m) be an H-space. The multiplication m induces a binary operation on the set, [X,A], of based homotopy classes of maps of A into X. A classical result due to I. M. James [Q. J. Math. Oxford, II. Ser. 11, 161-179 (1960; Zbl 0097.161)] asserts that, moreover, [X,A] is an algebraic loop. That is, [A,X] has a two-sided identity element and if a,b\(\in [A,X]\) then the equations \(ax=b\) and \(ya=b\) have unique solutions x,y\(\in [A,X]\). This paper is a detailed study of the following loop theoretic notions applied to H-spaces: inversivity, power-associativity, quasi-commutativity and the Moufang property. In particular the authors answer the following natural questions for finite CW H-spaces: Are there any multiplications which are not inversible? Are there any multiplications which are not power- associative? Which H-spaces admit a non-quasi-commutative multiplication? Which H-spaces admit a non-Moufang multiplication? Their method consists first in reducing the study of a multiplication m on X to the study of the diagonal on the free commutative, associative algebra \(H^*(X_{{\mathbb{Q}}})\) induced by the rationalization \(m_{{\mathbb{Q}}}\) of m. Then, using the fact that rationalization preserves the loop-theoretic properties, they obtain conditions on X itself.