Strong Cayley's Theorem as stated in Topics in Algebra by I.N. Herstein :
If G is a group, H a subgroup of G, and S is the set of all right cosets of H in G, then there is a homomorphism $ \theta $ of G into A(S) and the kernel of $ \theta $ is the largest normal subgroup of G which is contained in H
*(note1: For any set $ S $ , we define $ A(S) $ as the set of all bijections on S)
*(note2: Mapping which is not onto is called an into mapping)
For the proof, please refer to page 85, 86 of the pdf
My question is: how can we say that the above mentioned homomorphism $ \theta $ is into?
Can anyone please explain?