# Whence “homomorphism”, “homomorphic”?

The kernel question leads to another : Today, homomorphism (resp. isomorphism) means what Jordan (1870) had called isomorphism (resp. holoedric isomorphism). How did the switch happen?

• “Homomorphic” appears e.g. in de Séguier (1904, pp. 65–66):

(...) supposons entre les groupes $\mathrm A$ et $\mathrm B$ une correspondance telle qu’à chaque élément de $\mathrm A$ réponde un élément au moins de $\mathrm B$ et à chaque élément de $\mathrm B$ un élément au moins de $\mathrm A$ et que si $a_i$ de $\mathrm A$ et $b_i$ de $\mathrm B$ se correspondent, $a_ia_k$ et $b_ib_k$ se correspondent aussi. On dit que $\mathrm A$ et $\mathrm B$ sont homomorphes.

• “Homomorphism” (as property of being homomorphic) appears e.g. in Loewy (1912, p. 198):

§1 (...) untersucht den Homomorphismus zweier abstrakter Gruppen (...).

• “Homomorphism” (as map between groups) appears e.g. in Hopf (1928, p. 128):

4) Unter einem Homomorphismus der Gruppe $\mathfrak G$ in die Gruppe $\mathfrak H$ verstehen wir eine eindeutige Abbildung $f$ von $\mathfrak G$ auf einen echten oder unechten Teil von $\mathfrak H$, bei der stets $f(x+y)=f(x)+f(y)$ ist.

Are these the earliest occurrences?

• The following text notes the use of homomorphism by de Séguier in 1904, and then states that the term was introduced by Klein without giving a reference. books.google.ca/… – Nick Aug 27 '17 at 17:31
• Homomorphism is etymologically correct in the sense that the structures of two groups exhibit a degree of similarity when one can map into the other (trivial cases apart). Isomorphism is then more accurately used to mean "congruent" structure. ("Isos" is more commonly used to mean equal, but Euclid also uses it for congruent.) – Tony Solomonides Aug 31 '17 at 18:32
• Now cross-posted: mathoverflow. – Francois Ziegler Sep 3 '17 at 17:53