What is the name of this theorem

Question

Let $f: G\to K$ a morphism of groups. If $H\subset \ker f$, then there existe a unique morphism of groups $g : G/H\to K$ such that $f=gs$. Moreover,

$g$ is surjective if $f$ is surjective ;

$g$ is injective if we have $H=\ker f$ ;

$g$ in an isomorphism if $f$ is surjetive and $H=\ker f$

http://fr.wikipedia.org/wiki/Th%C3%A9or%C3%A8me_de_factorisation#Le_cas_des_groupes

Thank you.

Answer 1

I would be tempted to call this the 1st isomorphism theorem of groups, although that is usually stated in the weaker form $G/\ker{f}\cong\operatorname{im}(f)$ (with $f$ as in your question).

To a degree I think the theorem you state is what should really be called the 1st isomorphism theorem, and I doubt it's different enough from what is usually called the 1st isomorphism theorem to have a new name.