In studying vector spaces we learn about linear transformations from one vector space to another and in particular the kernel of such a transformation. When learning about group theory we also learn of kernels of homomorphisms. I know that group theory generalises the idea of kernel to work with other groups, but I was wondering which came first... was the kernel originally defined from the study of vectors spaces or from the more general group theory?


2 Answers 2


According to an entry attributed to John Aldrich in Jeff Miller's Earliest Known Uses of Some of the Words of Mathematics, the word "kernel" in the algebraic sense was first used in the 1946 translation of Pontryagin's book Topological Groups by Emma Lehmer. So kernels of group homomorphisms came first. An unrelated use as in "integral kernel" occurs earlier, in Hilbert's 1904 Grundzüge einer Allgemeinen Theorie der Linearen Integralgleichungen, in German, and in Bochner's 1909 Introduction to the Study of Integral Equations, in English.

Here is a Mathematics Stack Exchange thread, that has speculations about Pontryagin's reasons for the name.

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    $\begingroup$ Even if not "active" Jeff Miller's site is still there: jeff560.tripod.com/mathword.html $\endgroup$ Apr 11, 2015 at 21:52
  • $\begingroup$ @Gerald Edgar I saw this link but when I click on it I always get "Problem loading page". $\endgroup$
    – Conifold
    Apr 12, 2015 at 19:23
  • $\begingroup$ The page works fine for me. But it has been a year or so since the last update. Maybe those "first uses" don't keep changing... $\endgroup$ Apr 12, 2015 at 19:27
  • $\begingroup$ jeff560.tripod.com/c.html had revisons in July 2017. $\endgroup$ Aug 26, 2017 at 14:32

The quoted 1946 English edition of Pontryagin's 1938 book is not the first appearance of kernel.

  1. It's already in the first English edition (1939).

  2. It's earlier in e.g. 1938 papers of J. H. C. Whitehead (p. 703) and Montgomery-Zippin (p. 364).

  3. The original German Kern is in P. Alexandroff and H. Hopf, Topologie I (1935, p. 557), and as Jan Peter comments below, in Pontryagin (1931, pp. 186, 189, 202) – with p. 186 sounding like he’s introducing the word:

    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.

(Still, it seems to have come up in group theory first, and only later in linear algebra or ring theory.)

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    $\begingroup$ Pontryagin used the term "Kern des Homomorphismus" already in his 1931 paper in "Math. Annalen 105" $\endgroup$ Aug 26, 2017 at 8:02
  • $\begingroup$ @JanPeterSchäfermeyer Right! Page 186 even italicizes it like a definition. $\endgroup$ Aug 26, 2017 at 9:03

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