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I've always found that isomorphisms are more natural than homomorphism. Isomorphisms are motivated by the desire of defining when two structures are "structurally equal". But what is the motivation of "homomorphism"? In many presentations it seems that homomorphisms are motivated just as a stepping stone to the definition of "isomorphism".

But what is the real motivation of considering homomorphisms independently of isomorphisms? Why are people interested in studying these? Why do they help us reveal things about the structures we care about? What was historically the motivation?

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    The injective ones give you embeddings, and the surjective ones give you quotients. – Randall Sep 04 '19 at 13:12
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    Isomorphism is simply far too strong, two structures are only isomorphic if they are "basically the same" - while homomorphisms preserve some of the structure but not all giving us more possibilities. – Qi Zhu Sep 04 '19 at 13:13
  • See also https://math.stackexchange.com/questions/242348. I've seen similar questions very often already. – Qi Zhu Sep 04 '19 at 13:26
  • @QiZhu: The thread you linked to perfectly answers the question about what a homomorphism is intuitively, here I rather ask about the motivation for considering them. –  Sep 04 '19 at 13:28
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    I see plenty of motivation in those answers. – Randall Sep 04 '19 at 13:28
  • What is the exact difference between intuition and motivation for you? By understanding the intuition, you should hopefully also understand the motivation in this case. – Qi Zhu Sep 04 '19 at 13:30
  • @Randall: Can you point out the motivation? –  Sep 04 '19 at 13:53
  • @QiZhu: The difference is that intuition covers what the notion means, while motivation explains why we care. –  Sep 04 '19 at 13:54
  • @user7280899 yes: understanding embedded/quotient structure that may not occur from a strict iso. – Randall Sep 04 '19 at 13:56
  • @user7280899 also, you may want to consider the linear algebra point of view. Every matrix defines a homomorphism, but only the square ones of full rank define isos. Are you ready to throw away all matrices that aren't invertible? – Randall Sep 04 '19 at 14:07
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    @user7280899 Thank you, that's also how I would distinguish these words. However, as I've already mentioned, in this case you should understand the motivation if you get the intuition - it is simply such a natural concept. – Qi Zhu Sep 04 '19 at 14:19
  • Basically most has already been covered by what Randall and I have said as well. In mathematics, we are mostly interested in functions (see also category theory!) because they encode much of the information of the objects (see again category theory!). When we're dealing with groups however, we do not only want "usual functions" but rather special functions that are somehow connected with groups. They should at least preserve some structure to give us some information about groups. And the most natural such map is simply a group homomorphism. – Qi Zhu Sep 04 '19 at 14:22
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    Notice also that Randall suggests you to think of linear maps. This is no coincidence because in every part of mathematics there are morphisms/structure-preserving maps. Again, category theory formalizes this but the main takeaway should be this: What we're interested in in mathematics are not the objects but the maps. – Qi Zhu Sep 04 '19 at 14:25
  • @QiZhu: I feel that I perfectly understand what a homomorphism is. Yet I am curious about the motivation. –  Sep 04 '19 at 17:28
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    Did you not read my two latter comments? I tried to explain the motivation to you... – Qi Zhu Sep 04 '19 at 22:19
  • @QiZhu: Could you be more specific? How are homomorphisms used to do things? –  Sep 05 '19 at 08:24
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    Your entire course on algebra should be about this. – Qi Zhu Sep 05 '19 at 13:57

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