Let $S1$ and $S2$ be two sets. Suppose that there exists a one-to-one mapping $J$ of $S1$ into $S2$ . Show that there exists an isomorphism of $A(S1)$ into $A(S2)$, where $A(S)$ means the set of all one-to-one mappings of $S$ onto itself.
I am not able to find the homomorphic map because $J$ is not necessarily onto.If $J$ was onto we have define a map in which each symbol in an element $x$ belonging to $A(S1)$ could be replaced by corresponding in S2 by using the map $J$.
This Question is from Herstein 2.7.21 .