<p>This paper investigates the indefinite optimal control and stabilization of discrete-time mean-field stochastic Markov jump system. The necessary and sufficient conditions for the solvability of the optimal control problem with finite horizon, as well as the optimal controller and performance index, are derived through the application of a novel maximum principle and the corresponding generalized coupled difference Riccati equation. For the infinite horizon case, under the exact observability of the system, if certain linear matrix inequalities are satisfied and the sets of kernel constraint are non-empty, the necessary and sufficient conditions for mean square stabilization of this system, along with the associated optimal controller, are obtained from a novel Lyapunov function and the generalized coupled algebraic Riccati equation. The discrete-time mean-field stochastic Markov jump system with indefinite weight costs is mean square stabilizable, if and only if the algebraic Riccati equation has a unique maximum solution. Moreover, the proposed stable condition in this paper is novel, and the effectiveness of the algorithm will be validated through numerical simulations.</p>