Notice the similarity but also the difference to the definition of the centralizer subgroup, for which in the above.
If the subset is in fact a subgroup of , then it is a normal subgroup of the normalizer ; and is the largest subgroup of such that is a normal subgroup of it, whence the terminology normalizer. Indeed, if is already a normal subgroup of , then its normalizer coincides with the whole of .
Hence when is a group then the quotient
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 .
In (Gray 14) the concept of the normalizer of a subgroup of a group is generalized to the normalizer of a monomorphism in any pointed category in terms of a universal decomposition of a monomorphism with a normal monomorphism.
In (Bourn-Gray 13) the condition that be a monomorphism is dropped.
James Richard Andrew Gray, Normalizers, centralizers and action representability in semiabelian categories, Applied Categorical Structures 22(5-6), 981–1007, 2014.
Peter May, Equivariant homotopy and cohomology theory CBMS Regional Conference Series in Mathematics, vol. 91, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1996. With contributions by M. Cole, G. Comezana, S. Costenoble, A. D. Elmenddorf, J. P. C. Greenlees, L. G. Lewis, Jr., R. J. Piacenza, G. Triantafillou, and S. Waner. (pdf)