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Let KG denote the group algebra of the group G over a field K of characteristic 0. The weak (homological) dimension of KG is defined as the largest integer q for which there exist KG-modules B, C such that Torq G(B, C) 0. By [l; p. 352] q is equal to the largest integer for which there exists a KG-module A such that Hq(G, A) 0.We write w.dim KG = q. The following is a corollary of the main result (Theorem 1): For a solvable group G the weak dimension of KG is equal to the Hirsch number hG of G, w.dim KG = hG.As an application we obtain (Theorem 5) : If G is a nilpotent group of finite Hirsch number hG, then hG is equal to the largest integer q for which Hq (G, K + ) 0 0, where K + denotes the additive group of K with trivial G-operation.The starting point of our investigation was a result of K. W. Gruenberg [3], communicated privately to the author, which relates the Hirsch number of a torsionfree nilpotent group to the cohomological dimension of the group. We would like to express our sincere thanks to Gruenberg for showing us his result. Definition. Let G be a group in C, and {Ni} a series of normal subgroups with the above property. We define the Hirsch number hG of G to be the sum of the ranks of those quotients N^/N i+ 1 which are abelian. If this sum is not finite, we write hG = oo. The Hirsch number of G and the weak dimension of KGThe fact that hG does not depend on the series of subgroups is well known; also, it follows immediately from our Theorem 1. (In the literature the Hirsch number is usually defined for a slightly different class of groups, see for example [4; p. 150]. However, the above definition is better for our purpose.) THEOREM 1 . For G in C we have hG = w.dim KG.We first prove a special case: PxorosiTtorr 2 [2]. Let N be an abelian group of rank n. Then w.dim KN = n and H^(N, K + ) = K.Proof. We proceed by induction on n.