stabilizer subgroup

Said of a group acting on a set, and with respect to an element of that set: the subgroup of the given group whose action leaves that element fixed.

Said of a group acting on a set, and with respect to a subset of that set: the subgroup of the given group for which the image of the set under the group action is a subset of the original set. Some authors instead define this to be the subgroup of the given group whose action leaves each element of the subset fixed.