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Several elementary properties of the symmetric group $S_n$ extend in a niceway 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 somesense the torsion part of the monoid $M_n$. More precisely, there is apretorsion theory in the category of all maps $f\colon X\to X$, $X$ anarbitrary finite non-empty set, in which bijections are exactly the torsionobjects.

An extension of properties of symmetric group to monoids and a pretorsion theory in the category of mappings