This paper considers a linear-quadratic (LQ) mean field control problem involving a major agent and a large number of minor agents. The objective is to optimize a social cost as a weighted sum of the individual costs under decentralized information, and so the situation may be termed a mean field team problem. We apply the person-by-person optimality principle in team decision theory to the finite population model to construct two limiting optimal control problems whose solutions, subject to the requirement of consistent mean field approximations, yield a system of forward-backward stochastic differential equations (FBSDEs). We show the existence and uniqueness of a solution to the FBSDEs and obtain decentralized strategies nearly achieving social optimality in the original large but finite population model.