The following is a simple proof of why .

For instance . Note that the tricky part is that is not actually the usual {0,1}, but rather {0,3} (considered as part of ). Hence the elements of are {0,3}, {1, 4}, {2, 5}, which can be seen to be isomorphic to .

A sketch of a proof is as follows. Consider , where , defined by .

We may check that it is well-defined since if , then , and thus .

It is a fairly straightforward to check it is a homomorphism,

Injectivity is clear since , and surjectivity is quite clear too.

Hence, this ends the proof. 🙂

Do check out some Recommended Books on Undergraduate Mathematics, and also download the free SG50 Scientific Pioneers Ebook, if you haven’t already.

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