Jordan–Hölder theorem can be used to prove the fundamental theorem of arithmetic. But I can only prove the uniqueness part of the theorem with Jordan–Hölder theorem. That every composite number is expressible as a product of primes cant be proved. So how can I complete my proof?
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asked Nov 16, 2016 at 16:40
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$\begingroup$ I assume that you are applying Jordan-Holder to $\mathbb Z / n\mathbb Z$? Since the theorem gives that the factors of the composition series are unique, all you have to do is show that they are of the form $\mathbb Z/p\mathbb Z$ for $p$ prime. But assume that some factor is not irreducible. What happens? $\endgroup$– AaronCommented Nov 16, 2016 at 16:53
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$\begingroup$ This follows almost by definition of composite, not from the Jordan Holder Theorem. If $n$ is composite, then $n = n_1 n_2$. Then check whether each of $n_1, n_2$ are prime. $\endgroup$– NitinCommented Nov 16, 2016 at 16:55
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