Editing Homomorphism (section)

{{Short description|Structure-preserving map between two algebraic structures of the same type}}
{{Distinguish|Holomorphism|Homeomorphism}}
In [[algebra]], a '''homomorphism''' is a [[morphism|structure-preserving]] [[map (mathematics)|map]] between two [[algebraic structure]]s of the same type (such as two [[group (mathematics)|group]]s, two [[ring (mathematics)|ring]]s, or two [[vector space]]s). The word ''homomorphism'' comes from the [[Ancient Greek language]]: {{wikt-lang|grc|ὁμός}} ({{transliteration|grc|homos}}) meaning "same" and {{wikt-lang|grc|μορφή}} ({{transliteration|grc|morphe}}) meaning "form" or "shape". However, the word was apparently introduced to mathematics due to a (mis)translation of German {{wikt-lang|de|ähnlich}} meaning "similar" to {{lang|grc|ὁμός}} meaning "same".<ref>{{Cite book|last=Fricke|first=Robert|url=https://archive.org/details/vorlesungenber01fricuoft/page/n5/mode/2up|title=Vorlesungen über die Theorie der automorphen Functionen|language=de|date=1897–1912|publisher=B. G. Teubner|oclc=29857037}}</ref> The term "homomorphism" appeared as early as 1892, when it was attributed to the German mathematician [[Felix Klein]] (1849–1925).<ref>See:
*  {{cite journal |last1=Ritter |first1=Ernst |title=Die eindeutigen automorphen Formen vom Geschlecht Null, eine Revision und Erweiterung der Poincaré'schen Sätze |language=de |journal=Mathematische Annalen |date=1892 |volume=41 |pages=1–82 |doi=10.1007/BF01443449 |s2cid=121524108 |url=https://babel.hathitrust.org/cgi/pt?id=hvd.32044102918109&view=1up&seq=15 |trans-title=The unique automorphic forms of genus zero, a revision and extension of Poincaré's theorem |quote=[footnote p. 22:] Ich will nach einem Vorschlage von Hrn. Prof. Klein statt der umständlichen und nicht immer ausreichenden Bezeichnungen: 'holoedrisch, bezw. hemiedrisch u.s.w. isomorph' die Benennung 'isomorph' auf den Fall des ''holoedrischen'' Isomorphismus zweier Gruppen einschränken, sonst aber von 'Homomorphismus' sprechen, ...|trans-quote=Following a suggestion of Prof. Klein, instead of the cumbersome and not always satisfactory designations "holohedric, or hemihedric, etc. isomorphic", I will limit the denomination "isomorphic" to the case of a ''holohedric'' isomorphism of two groups; otherwise, however, [I will] speak of a "homomorphism", ...}}
*  {{cite journal |last1=Fricke |first1=Robert |title=Ueber den arithmetischen Charakter der zu den Verzweigungen (2,3,7) und (2,4,7) gehörenden Dreiecksfunctionen |language=de |journal=Mathematische Annalen |date=1892 |volume=41 |issue=3 |pages=443–468 |doi=10.1007/BF01443421 |s2cid=120022176 |url=https://babel.hathitrust.org/cgi/pt?id=hvd.32044102918109&view=1up&seq=471 |trans-title=On the arithmetic character of the triangle functions belonging to the branch points (2,3,7) and (2,4,7) |quote=[p. 466] Hierdurch ist, wie man sofort überblickt, eine homomorphe*) Beziehung der Gruppe Γ<sub>(63)</sub> auf die Gruppe der mod. n incongruenten Substitutionen mit rationalen ganzen Coefficienten der Determinante 1 begründet. ... *) Im Anschluss an einen von Hrn. Klein bei seinen neueren Vorlesungen eingeführten Brauch schreibe ich an Stelle der bisherigen Bezeichnung 'meroedrischer Isomorphismus' die sinngemässere 'Homomorphismus'.|trans-quote=Thus, as one immediately sees, a homomorphic relation of the group Γ<sub>(63)</sub> is based on the group of modulo n incongruent substitutions with rational whole coefficients of the determinant 1. ... Following a usage that has been introduced by Mr. Klein during his more recent lectures, I write in place of the earlier designation 'merohedral isomorphism' the more logical 'homomorphism'.}}</ref>

Homomorphisms of vector spaces are also called [[linear map]]s, and their study is the subject of [[linear algebra]].

The concept of homomorphism has been generalized, under the name of [[morphism]], to many other structures that either do not have an underlying set, or are not algebraic. This generalization is the starting point of [[category theory]].

A homomorphism may also be an [[isomorphism]], an [[endomorphism]], an [[automorphism]], etc. (see below). Each of those can be defined in a way that may be generalized to any class of morphisms.