I am trying to prove that these 2 groups are homomorphic first, since I know that isomorphic groups are isomorphic if they are homomorphic and bijective. My problem is that I don't know which operation to use when proving homomorphism.

  • $\begingroup$ Can you think of a function that can turn multiplication into addition? And historically was used for precisely that reason - to simplify multiplication by reducing it to addition? Used in slide rules. $\endgroup$ – user491874 Nov 12 '17 at 17:08
  • $\begingroup$ The Log function? $\endgroup$ – animorphlover3 Nov 12 '17 at 17:08
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    $\begingroup$ Correct, and I can see your question has in the meantime been answered with the same answer. $\endgroup$ – user491874 Nov 12 '17 at 17:09
  • $\begingroup$ What does homomorphic mean? $\endgroup$ – Derek Holt Nov 12 '17 at 17:26

$\ln(xy) = \ln(x) + \ln(y)$ shows that the natural logarithm is a homomorphism from the multiplicative group of reals to the additive group. You also need to check that it is one-to-one and onto, to conclude that it is an isomorphism.

As for how you would come up with this operation, I'm not sure that everyone would necessarily come up with this answer on their own. It may be something that was covered in lecture.

  • $\begingroup$ Thank you! That makes sense! $\endgroup$ – animorphlover3 Nov 12 '17 at 17:10
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    $\begingroup$ @animorphlover3 FYI it can be logarithm with any base. Common logarithm if you prefer. $\endgroup$ – ziggurism Nov 12 '17 at 17:11

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