This previous question traces the notion of group homomorphism to Jordan (1870) and the term "homomorphic" to Fricke and Klein (1897) and to earlier lectures of Klein:


[Whence “homomorphism” and “homomorphic”?](https://mathoverflow.net/questions/280261/whence-homomorphism-and-homomorphic)


What about the concept of the **kernel** of a homomorphism?


This webpage traces it to Fredholm's (1903) use of the french "noyau" for the nullspace of a system of integer linear equations. Then Hilbert put this into German as "Kern" (1904).


<http://jeff560.tripod.com/k.html>


But the concept of a kernel is structural, equivalent to the concept of normal subgroup. I guess that this insight is due to **Emmy Noether** but I can't find a reference.


Probably what I am looking for is the first enunciation of the theorem $$G/\ker\varphi\cong\mathrm{im}\,\varphi$$.



History of the kernel of a homomorphism?
The *word* at least, seems to originate with **Pontryagin** ([1931](//zbmath.org/?q=an:57.0717.01), [p. 186](//hsm.stackexchange.com/a/5867)):



> 
> 28) Wenn eine Gruppe $A$ auf eine Gruppe $B$ homomorph abgebildet ist, so heißt die Untergruppe von $A$, die aus allen Elementen besteht, welche auf das Einheits- (Null-) element von $B$ abgebildet werden, der ***Kern*** der homomorphen Abbildung.
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E.g. van der Waerden ([1930](//zbmath.org/?q=an:56.0138.01), [p. 35](//google.com/books?id=8DKbBwAAQBAJ&pg=PA35)) still states the isomorphism $G/\ker\varphi\cong\mathrm{im}\,\varphi$ without $“\ker”$, as $\mathfrak{G/e\cong\overline{G}}$ with $\mathfrak e$: *derjenige Normalteiler von $\mathfrak G$, dessen Elementen das Einselement in $\overline{\mathfrak G}$ entspricht*.




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**Added:** Noether wrote things like $G/\varphi^{-1}(H')\cong G'/H'$, but I’m not sure she ever spelled out the case $H'=\{e\}$. Anyway, the theorems whose “wording and proof” van der Waerden ([1975](//ams.org/mathscinet-getitem?mr=465721), p. 34) attributes to Noether are those for “groups with operators” ([1930](//zbmath.org/?q=an:56.0138.01), [p. 136](//google.com/books?id=8DKbBwAAQBAJ&pg=PA136)). For groups, he says he followed Speiser, who has indeed ([1923](//zbmath.org/?q=an:49.0081.01), [p. 19](//google.com/books?id=tMHVAAAAMAAJ&pg=PA19)):



> 
> **Satz 14:** *Ist $\mathfrak G'$ mit $\mathfrak G$ isomorph, so entspricht dem Einheitselement von $\mathfrak G'$ ein Normalteiler $\mathfrak N$ von $\mathfrak G$, und $\mathfrak G'$ ist homomorph mit der Faktorgruppe $\mathfrak{G/N}$*.
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and Burnside, who has ([1897](//zbmath.org/?q=an:28.0118.03), [pp. 36, 38](//archive.org/details/117768378/page/36)):



> 
> **Theorem VII.** *If a group $G$ is multiply isomorphic with a group $G'$, then* (i) *the operations of $G$, which correspond to the identical operation of $G'$, form a self-conjugate sub-group of $G$;*
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> 
> (...)
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> 
> a group $G'$ with which a group $G$ is multiply isomorphic, in such a way that to the identical operation of $G'$ there corresponds a given self-conjugate sub-group $\Gamma$ of $G$, is completely defined (as an abstract group) when $G$ and $\Gamma$ are given. (...) Herr Hölder ([1889](//zbmath.org/?q=an:21.0078.01), p. 31) has introduced the symbol
>  $
> \smash{\frac G\Gamma}
> $
>  to represent this group; he calls it the *quotient* of $G$ by $\Gamma$, and a *factor-group* of $G$.
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I think it’s fair to say that Hölder (pp. 32–33) already has all of the above, except the word *kernel*. (E. g. he writes that normal subgroups of $G|\mathsf H$ *make* normal subgroups of $G$, with the identity *making* $\mathsf H$ itself, and that one could *start* from a morphism rather than a normal $\mathsf H$.) But it goes further: kernels are already in Dyck ([1882](//zbmath.org/?q=an:14.0097.01), p. 12, cited by Hölder):



> 
> *Operationen der Gruppe $G$, welche sonach der Identität in $\overline G$ entsprechen, bilden eine Gruppe $H$ und diese ist (...) in $G$ ausgezeichnet enthalten.*
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in Capelli ([1878](//zbmath.org/?q=an:10.0105.03), [p. 36](//books.google.com/books?id=GD0FAAAAYAAJ&pg=PA36)), who calls them *primi periodi* (denoted $\mathrm O\_0$) and shows:



> 
> *Affinchè un gruppo $\mathrm O$ contenuto in $\mathrm G$ possa esser preso come primo periodo, à necessario che esso si permutabile a tutte le sostituzioni di $\mathrm G$.* Vedremo più tardi (III, 3) che questa condizione à anche sufficiente, vale a dire, che si può sempre costruire un gruppo $\Gamma$ isomorfo a $\mathrm G$, il quale ad $\mathrm O$ faccia corrispondere l’unità.
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in Jordan ([1870](//zbmath.org/?q=an:03.0042.02), [§67](//archive.org/details/traitdessubsti00jorduoft/page/56), cited by Capelli):



> 
> Le groupe $\Gamma$ contient la substitution $ı$. Soient $h\_1,\dots,h\_m$ les substitutions correspondantes de $\mathrm G$ : elles forment un groupe auquel toutes les substitutions de $\mathrm G$ sont permutables.
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and finally(?), as per Noether’s *“[Es steht alles schon bei Dedekind](//ams.org/mathscinet-getitem?mr=465721)”*, in **Dedekind** ([1855–58](//zbmath.org/?q=an:58.0042.09), [p. 440](//gdz.sub.uni-goettingen.de/id/PPN23569441X?tify=%7B"pages":[443]%7D)), for a morphism $M\to M\_1$:



> 
> Der Komplex aller der $n$ in $M$ enthaltenen Objekte $\varphi$, denen das Objekt $1$ entspricht, bildet eine Gruppe, und zwar einen *eigentlichen Divisor* von $M$; dann ist $m = m\_1n$. 
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