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The question is that simple, Why isomorphisms are important ?

I know how we define mathematically isomorphisms, and I know what it means, if two things are isomorphic they are essentially the same thing. I want you to give some examples where they answer the question

"Why isomorphisms are important ?" because I am an undergraduate I don't have that much experience to give my self that example. I think isomorphism like the analog of matrix similarity, meaning if we want to find a big power of a matrix we don't calculate all the matrix multiplications we just find the Jordan form or the diagonal matrix, and we take it from here. So we make problems easier and faster to calculate by connecting them with isomorphisms. Do you have in mind a fundamental example? Or one you think condenses the beauty of mathematics.

Thank you.

José Carlos Santos
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    Perhaps an example is helpful. Consider the cyclic group $(\Bbb Z/3\Bbb Z,+)$. There are infinitely many different ways to describe this group. But the properties of all these "different" groups are the same. There is only one such group up to isomorphism. That's why isomorphisms are important. – Dietrich Burde Apr 20 '21 at 11:18
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    If two things are isomorphic then they are the same thing with different names. It is analogous to you and I using a different variable name for the same object. – John Douma Apr 20 '21 at 13:44
  • @JohnDouma (I'm assuming this is a joke) if it's not did you read my question ? I asked for a fundamental example where we're using that. – 領域展開 Apr 20 '21 at 15:46
  • This is not a joke. Every isomorphism you know of has this property. Your question shows a lack of insight. I gave it the attention it deserved. – John Douma Apr 20 '21 at 16:41
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    I don't think you read my question, in my question I said if two things are isomorphic they are essentially the same thing then i brink the example of Jordan forms, your comment said If two things are isomorphic then they are the same thing with different names. Exactly how was that helpful? I don't find a reason writing that comment. – 領域展開 Apr 20 '21 at 18:18
  • You are like a child who says something profound but doesn't understand what he said. That isomorphism means sameness is why they are important. – John Douma Apr 20 '21 at 21:17

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