I came across two interesting articles relating to the Jordan-Holder Theorem:
The first gives a proof of the theorem, and the second shows how one can use it to prove the Fundamental Theorem of Arithmetic. My two questions are:
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In the proof of Jordan-Holder, how does the second theorem of isomorphisms show that $L$ is a maximal subgroup of $H$?
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In the proof of the Fundamental Theorem of Arithmetic, it is not verified that the groups $H_i$ are such that $H_{i+1}\triangleright H_i$. In this case we have groups such as $\mathbb{Z}/n\mathbb{Z}$ and $\mathbb{Z}/(n/p_i)\mathbb{Z}$. Can anyone show that the latter of these two groups is normal in the former? I have never seen the use of something like $(n/p_i)$ before in such an expression, so a little explanation would be much appreciated. How exactly would one describe the group $\mathbb{Z}/\mathbb{Z}$?
Best Answer
To your first question use the fact: $A$ is maximal proper normal subgroup of $B$ $\Leftrightarrow$ $B/A$ is simple.
To your second question since $\mathbb{Z}/n\mathbb{Z}$ is abelian every subgroup is normal and therefore $\mathbb{Z}/(n/p_i)\mathbb{Z}$ is a normal subgroup of $\mathbb{Z}/n\mathbb{Z}$. $(n/p_i)$ means $n$ divided by $p_i$.