# History of the kernel of a homomorphism?

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”?

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 $$G/\ker\varphi\cong\mathrm{im}\,\varphi$$.

The word at least, seems to originate with Pontryagin (1931, p. 186):

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.

E.g. van der Waerden (1930, p. 35) 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.

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, p. 34) attributes to Noether are those for “groups with operators” (1930, p. 136). For groups, he says he followed Speiser, who has indeed (1923, p. 19):

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 $$\mathfrak{G/N}$$.

and Burnside, who has (1897, pp. 36, 38):

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 $$G$$;

(...)

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, 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 $$G$$.

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, 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.

in Capelli (1878, p. 36), 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.

in Jordan (1870, §67, 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.

and finally(?), as per Noether’s Es steht alles schon bei Dedekind, in Dedekind (1855–58, p. 440), 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 $$m = m_1n$$.

Two original references are:

E. Noether, Hyperkomplexe Größen und Darstellungstheorien (1929): see page 648.

E. Noether, Abstrakter Aufbau der Idealtheorie in algebraischen Zahl- und Funktionenkörpern (1927): see page 40.

Noether phrases the isomorphism theorems in terms of "Normalteiler" (normal subgroups) rather than "Kerne" (kernels). Van der Waerden recalls that the 1929 paper was based on a course with the same title that Noether taught at Göttingen in 1926/1927.