In this paper, we develop a social group utility maximization game model that takes into account both social relationships and physical coupling among users. Specifically, instead of maximizing one's individual utility, each user aims to maximize its social group utility that hinges heavily on its social ties with other users. A salient feature of this model is that it spans the continuum space between non-cooperative game and network utility maximization - two extreme paradigms based on drastically different assumptions that users are selfish and altruistic, respectively. Based on this model, we study two important applications in mobile social networks: random access control and power control, and quantify the impact of social ties on users' strategies and network efficiency. In particular, our results show that, as the strength of social ties increases from the minimum to the maximum, the social-aware Nash equilibrium strategy of a player in this model migrates from the Nash equilibrium strategy in a standard non-cooperative game to the social-optimal strategy in network utility maximization. Therefore, the proposed social group utility maximization game model offers a general framework that encompasses non-cooperative game and network utility maximization as special cases, and we believe that it will open a new door to exploring the impact of social behavior on networking.