A value function for cooperative games with transferable utility is a function which assigns to every such a game a distribution of the payoffs over the players. An alternative type of solutions are share functions which assign to every player its share in the payoffs to be distributed. In this paper we consider cooperative games in which the players are organized into an a priori coalition structure being a finite partition of the player set. We introduce a general method for defining share functions for such games using a multiplication property that states that the share of a player in the total payoff is equal to its share in some internal game within its a priori coalition, multiplied by the share of this coalition in an external game between the a priori given coalitions. We provide axiomatizations of these coalition structure share functions using this multiplication and certain consistency properties. © 2004 Elsevier Inc. All rights reserved.