Abstract: This paper focuses on the stability of the 3D anisotropic incompressible magnetohydrodynamic (MHD) system with
 partial and fractional dissipation near a background magnetic field. We obtain the global well-posedness of solution, provided that the $H^2(\mathbb R^3)$ norm of initial datum is sufficiently small. Due to the emergence of fractional dissipation, some conventional energy estimate methods are no longer applicable.~To overcome this challenge, we establish new fractional anisotropic inequalities, which play a crucial role in bounding the nonlinear terms by the dissipative parts. Our result extends some existing conclusions.

Key words: 3D MHD system, partial and fractional dissipation, stability