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I read that: "The term "homomorphism" appeared as early as 1892, when it was attributed to the German mathematician Felix Klein (1849–1925). Homomorphisms of vector spaces are also called linear maps, and their study is the subject of linear algebra". (https://en.m.wikipedia.org/wiki/Homomorphism)

But definitely the concept have already been made (?). Because we know most of mathematicians of pre Klein era need this concept for their works.

I want to know who was he? And what was his inspiration and why did he introduce the term like this: f(xy)=f(x)f(y). And do we have a previous definition for homomorphism (or isomorphism or every other morphism)?

Iman Mosleh
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    "Because we know most of mathematicians of pre Klein era need this concept for their works"? We know this how? Just because homomorphisms are used to present their work today does not mean that they used the concept themselves. It requires a level of abstraction that was not reached until the end of 19th century. Gallian attributes the term to Jordan (1870), see Math SE, What is the history of the homomorphism concept? – Conifold Mar 08 '24 at 08:13
  • @Conifold So, What was pre Klein era version of homomorphisms? How did they use the concept? How did they relate two morphisms? I specially want to see the way they used to go without this definition of homomorphism. – Iman Mosleh Mar 08 '24 at 08:19
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    Jordan’s Traité is surveyed in Gray, History of Abstract Algebra, ch. 13. You can read what came before in previous chapters. They defined specific maps and used their properties without abstracting and packaging them into a concept. – Conifold Mar 08 '24 at 08:28
  • @Conifold "They defined specific maps and used their properties without abstracting and packaging them into a concept" It was almost good answer, Thanks – Iman Mosleh Mar 08 '24 at 08:54
  • Google ngram shows some use of the word 1855-1865, then another bump in 1880-1887, and since 1926 it comes to the common use. Does this word have some other meaning (in other sciences)? – Alexandre Eremenko Mar 08 '24 at 13:15
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    @AlexandreEremenko Yes. Collins has "homomorph biology something that is similar in form to something else" they also list homomorphic, homomorphosis with biological meanings. – Gerald Edgar Mar 08 '24 at 13:21
  • @Gerald Edgar: this does not explain the 19 century bumps that we observe: enter "homomorphism,homomorph" in the ngram. Well, a combination "homomorphism,homomorphic" gives some correlation. – Alexandre Eremenko Mar 08 '24 at 13:24
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    In an Ngram result, click at the bottom to get examples of the usage from various times. For example, we find "homomorphsm" in Proceedings of the Cambridge Philosophical Society but it is listed with the date 1843, which is when the journal started publication. – Gerald Edgar Mar 08 '24 at 13:25

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