Notation $S\twoheadrightarrow \,\,\,\stackrel{S}{}\!\!\unicode{x2215}_{\!\unicode{x202f}\sim} \hookrightarrow T$ in Group Theory

Question

I am learning group theory now. My professor wrote down some notation that I was not able to understand. Can you tell me the meanings?

$$S\twoheadrightarrow \,\,\,\stackrel{S}{}\!\!\unicode{x2215}_{\!\unicode{x202f}\sim} \hookrightarrow T$$

$$a\mapsto [a]\mapsto f(a)$$

Answer 1

$\twoheadrightarrow$ typically denotes surjection: so $f: A \twoheadrightarrow B$ means that $f$ is onto.

Dually, $\hookrightarrow$ indiciates injection. Sometimes it is used in a stricter sense to denote an honest inclusion of a subset. So if $A \subseteq B$ one may write $i: A \hookrightarrow B$ for the inclusion map $i(a)=a$.

$\mapsto$ indicates the action of a function on an element of the domain. So if $f: A \to B$ does $f(w)=z$, one may write $f: w \mapsto z$.

Also: this notation is not specific to group theory, but applies to functions between sets generally.