Notation for isomorphic groups: $A\cong B$ versus $A\xrightarrow{\sim} B$.

Question

If I have two isomorphic groups, can I write $A \xrightarrow{\sim} B$ rather than $A \cong B$ to mean "A is isomorphic to B", or is the arrow notation only used if I have a map $\varphi : A \xrightarrow{\sim} B$ ?

Answer 1

Upgraded to an answer from a comment by request:

I think it's better practice to use $A \cong B$ when you mean "there exists an isomorphism from A to B" and $A \xrightarrow{\sim} B$ when you mean "I have a specific isomorphism from A to B in mind". It's fine to use $A \cong B$ even in the latter case, but it would be strange to read $A \xrightarrow{\sim} B$ when there's no specific map being discussed.

Answer 2

If $A$ and $B$ are isomorphic, then it implies the existence of an isomorphism from one to the other. Though, that said, I wouldn't use that notation since it seems to imply you're talking about an isomorphism rather than the fact they're isomorphic.

Answer 3

Both notations are accepted.

If you're using either of them, though, it is good practise to state what it means; it is usually left as implicit from the context.

Having said that, it should be noted that "$A\stackrel{\sim}{\to}B$" usually refers to a specific isomorphism (from $A$ to $B$).