If is itself a subgroup, then is a normal subgroup of ; moreover is the largest subgroup of such that is a normal subgroup of it. Of course, if is itself a normal subgroup of , then its normalizer coincides with the whole of .
Each group embeds into the symmetric group on the underlying set of by the left regular representation where . The image is isomorphic to (that is, the left regular representation of a discrete group is faithful). The normalizer of the image of in is called the holomorph. This solves the elementary problem of embedding a group into a bigger group in which every automorphism of is obtained by restricting (to ) an inner automorphism of that fixes as a subset of .