In a school choice setting, this paper observes that no group strategy-proof mechanism satisfies even a fairly weak notion of stability. In response to this result, we introduce two monotonicity axioms, which we call top-dropping monotonicity and extension monotonicity, as alternatives to group strategy-proofness. We prove that these two axioms are equivalent to requirements that no group of students gains from a simple manipulation of their preferences. Then, replacing group strategy-proofness with the two axioms, we find that the Kesten’s (2010) efficiency adjusted deferred acceptance mechanism is the unique mechanism that satisfies the three criteria. We also provide several applications of the two monotonicity axioms especially for the deferred acceptance mechanism.