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Today homomorphism (resp. isomorphism) means what Jordan (1870) had called isomorphism (resp. holoedric isomorphism). How did the switch happen?

“Homomorphic” (and “homomorphism” as “property of being homomorphic”) are e.g. in de Séguier (1904, pp. 65–66) and the last edition of Weber (1912, p. 195). “Homomorphism” as map of groups is e.g. in Schur (1924, p. 191). But none of these sound like a first.

I asked this on hsm a week ago, but got no answer there.

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    $\begingroup$ Remark: the earliest uses site says 1935, but even in English this is predated by Pontrjagin (1934, p. 362), Tucker (1933, p. 196), and for algebra homomorphisms, Levitzki (1932, p. 382). $\endgroup$
    – Francois Ziegler
    Commented Sep 3, 2017 at 21:46
  • $\begingroup$ People were still using isomorphism onto and isomorphism into for surjective resp. injective homomorphisms, for far too long (IMHO). I've always found this weird, but I guess they were just not up to date with the change terminology. $\endgroup$
    – David Roberts ♦
    Commented Sep 3, 2017 at 21:52
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    $\begingroup$ @DavidRoberts Indeed, e.g., van der Waerden’s Moderne Algebra (1930, p. 32) still laments the absence of stable terminology. $\endgroup$
    – Francois Ziegler
    Commented Sep 3, 2017 at 22:11
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    $\begingroup$ Note that in the linked paper by Jordan, "group" means "permutation group", i.e., something which more or less means a set endowed with a subgroup of its group of permutations. Isomorphic means isomorphic as groups (not as permutation group). In particular when he says "Problem: determine all groups isomorphic to a given group $G$", it more or less means "classify all (faithful) actions of $G$ [the underlying group, in the modern sense]". $\endgroup$
    – YCor
    Commented Sep 3, 2017 at 22:14

1 Answer 1

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I found this footnote on page 195 of Fricke and Klein's Vorlesungen über die Theorie der automorphen Functionen (1897):

Translation:

The term "homomorphic" seems more appropriate than the previously$^\ast$ used "isomorphic", because it refers not to "equality" but to "similarity" of two groups. The term "isomorphism" will therefore from now on be used in the sense of "1-to-1 homomorphism".

$^*$l.c. = loco citato — this seems to refer to Fricke and Klein's Vorlesungen über die Theorie der elliptischen Modulfunctionen (1890).

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    $\begingroup$ Thank you. I now see that Frobenius (1902, p. 456) and de Séguier (1902, p. 257) also credit Felix Klein — despite Bourbaki (1981, p. VIII.462) attributing the word to Frobenius. $\endgroup$
    – Francois Ziegler
    Commented Sep 3, 2017 at 20:30
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    $\begingroup$ Moreover Frobenius points to Math. Ann. 41 (1892) where papers of (Klein’s student) Ritter and Fricke already make the switch, attributing it to Klein in his lectures: pp. 22, 466. $\endgroup$
    – Francois Ziegler
    Commented Sep 3, 2017 at 21:09
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    $\begingroup$ So "isomorphism" meant "injective homomorphism"? $\endgroup$
    – David Roberts ♦
    Commented Oct 5, 2021 at 23:15
  • $\begingroup$ @DavidRoberts Yes. Quoth Mac Lane (1988, p. 332): “At that time, a homomorphism in algebra always meant a surjective homomorphism (a mapping onto)” or (1970, p. 229): “For example, van der Waerden’s Moderne Algebra, following the lead of Emmy Noether, studies homomorphisms $G\to H$ of groups, and of rings, but only such as map $G$ onto $H$”... $\endgroup$
    – Francois Ziegler
    Commented Oct 28 at 1:27
  • $\begingroup$ ... Or indeed de Séguier (1904, p. 66): “supposons entre les groupes A et B une correspondance telle qu‘à chaque élément de A réponde un élément au moins de B et à chaque élément de B un élément au moins de A et que si $a_i$ de A et $b_i$ de B se correspondent, $a_ia_k$ et $b_ib_k$ se correspondent aussi. On dit que A et B sont homomorphes.” $\endgroup$
    – Francois Ziegler
    Commented Oct 28 at 1:27

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