A topological space with a continuous multiplication with unit is called an H-space. The topological properties of these spaces have been investigated by many authors, in particular the homology and homotopy groups. The case of Lie groups has been investigated intensely and many interesting results have been obtained by special methods for these groups. E. Cartan [8] proved that the second homotopy group of a Lie group is zero, a result which also follows from Bott's work [4]. In this paper we obtain a new proof of Cartan's theorem, using homological methods. Unlike the previous proofs which made strong use of the infinitesimal structure of Lie groups, the proof given here depends only on the homological structure and can be applied to H-spaces whose homology is finitely generated. If X is a simply connected H-space whose homology is finitely generated, then it follows from Hopf's theorem [12] on Hopf algebras that H2(X; R) = 0 (where R = real numbers) and hence that r2(X) is finite. The argument would show that a non-zero element x e H2(X; R) has infinite height (Xn # 0 for all n) which would contradict the hypothesis of finitely generated homology. Now if r2(X) # 0, then H2(X; Z.) # 0 for some prime p. While x e H2(X; Z,) may not in general have infinite height, a slightly weaker statement is proved; i.e., that x has a property called cc-implications (see definition in ? 6) which would again contradict the hypothesis that H*(X) is finitely generated. The above follows from a general theorem (Theorem 6.1) which gives a condition ensuring that an element will have cc-implications. Many consequences are deduced from this, particularly for H-spaces whose homology is finitely generated. If X is an arcwise connected H-space with H*(X) finitely generated, it is proved that the mod p Hurewicz homomorphism h: 7rm(X) 0 Z. ) Hm(X; Z.) is zero in even dimensions m, for all p, which implies that the first non-vanishing higher homotopy group of X occurs is an odd dimension. The known examples of H-spaces whose homology is finitely generated seem to consist of the Lie groups, the seven sphere S7, real projective 7-space P7 and products of these. These are all manifolds. It is shown here that for an H-space X with H*(X) finitely generated, the highest dimensional non-zero group Hn(X) is isomorphic to the integers Z, and * This paper was written while the author was a National Science Foundation Postdoctoral