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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)?

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    $\begingroup$ "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? $\endgroup$
    – Conifold
    Commented Mar 8 at 8:13
  • $\begingroup$ @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. $\endgroup$ Commented Mar 8 at 8:19
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    $\begingroup$ 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. $\endgroup$
    – Conifold
    Commented Mar 8 at 8:28
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    $\begingroup$ @AlexandreEremenko Yes. Collins has "homomorph biology something that is similar in form to something else" they also list homomorphic, homomorphosis with biological meanings. $\endgroup$ Commented Mar 8 at 13:21
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    $\begingroup$ 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. $\endgroup$ Commented Mar 8 at 13:25

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