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I'm trying to learn about groups/rings and the concept of isomorphisms appears everywhere. I understand that an isomorphism between groups/rings shows that arithmetic in both structures is essentially the same, but I don't understand how this can be applied to deduce anything about them. If someone could give an example or explain how isomorphisms can be used to answer a question I'd greatly appreciate it.

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    $\begingroup$ An example maybe is that the Mobius transformation is isomorphic to the projective linear group as in math.stackexchange.com/questions/1055559/…. This reduces asking about a geometric transformation to something that can be done with complex matrices, which is more concrete. $\endgroup$
    – twnly
    Dec 29, 2018 at 5:49

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