  Several elementary properties of the symmetric group $S_n$ extend in a nice
way to the full transformation monoid $M_n$ of all maps of the set
$X:=\{1,2,3,\dots,n\}$ into itself. The group $S_n$ turns out to be in some
sense the torsion part of the monoid $M_n$. More precisely, there is a
pretorsion theory in the category of all maps $f\colon X\to X$, $X$ an
arbitrary finite non-empty set, in which bijections are exactly the torsion
objects.
