# Composition series

Originally published at 狗和留美者不得入内. You can comment here or there.

My friend after some time in industry is back in school, currently taking graduate algebra. I was today looking at one of his homework and in particular, I thought about and worked out one of the problems, which is to prove the uniqueness part of the Jordan-Hölder theorem. Formally, if $G$ is a finite group and

$1 = N_0 \trianglelefteq N_1 \trianglelefteq \cdots \trianglelefteq N_r = G$ and $1 = N_0' \trianglelefteq N_1' \trianglelefteq \cdots \trianglelefteq N_s' = G$

are composition series of $G$, then $r = s$ and there exists $\sigma \in S_r$ and isomorphisms $N_{i+1} / N_i \cong N_{\sigma(i)+1} / N_{\sigma(i)}$.

Suppose WLOG that $s \geq r$ and as a base case $s = 2$. Then clearly, $s = r$ and if $N_1 \neq N_1'$, $N_1 \cap N_1' = 1$. $N_1 N_1' = G$ must hold as it is normal in $G$. Now, remember there is a theorem which states that if $H, K$ are normal subgroups of $G = HK$ with $H \cap K = 1$, then $G \cong H \times K$. (This follows from $(hkh^{-1})k^{-1} = h(kh^{-1}k^{-1})$, which shows the commutator to be the identity). Thus there are no other normal proper subgroups other than $H$ and $K$.

For the inductive step, take $H = N_{r-1} \cap N_{s-1}'$. By the second isomorphism theorem, $N_{r-1} / H \cong G / N_{s-1}'$. Take any composition series for $H$ to construct another for $G$ via $N_{r-1}$. This shows on application of the inductive hypothesis that $r = s$. One can do the same for $N_{s-1}'$. With both our composition series linked to two intermediary ones that differ only between $G$ and the common $H$ with factors swapped in between those two, our induction proof completes.

Tags:
• #### 中国姓氏的南北分布

有意思的是分布是很不均匀的，也就是说好多姓是南方比例远高于北方比例或相反。从这一点，你如果知道一个人的父母的姓氏，你就可以对他是北方人还是南方人有个有点统计显著的估计。 读者可以在包含大姓分布图的…

• #### Why I think Stanford professor James Landay's piece on computing in China in 2011 exposed his idiocy

I dittoed his blog post on my LiveJournal too, see https://gmachine1729.livejournal.com/175023.html. For the original, see…

• #### China Will Overtake the US in Computing…Maybe, Someday… by Stanford professor James Landay in 2011

Original at https://dubfuture.blogspot.com/2011/12/china-will-overtake-us-in.html. [note: the following is a rough draft — I appreciate…

• Post a new comment

#### Error

Anonymous comments are disabled in this journal

default userpic