Let G be a group with composition series. Let $H\triangleleft G$. Then $H$ is in some of the series.

Question

I know this is true for $G$ being finite. How about infinite group? Is this statement still valid?

I proved the finite case by induction on $|G|$ (which supposedly not valid for infinite case)

Could anyone give a hint or a counterexample?

Thanks!

Answer 1

Let's say that a group $G$ is poly-simple if it admits a Jordan-Hölder series (= a finite chain from $\{1\}$ to $G$ of subgroups, each normal in the next one, with simple successive quotients). This is the smallest class of groups, containing simple groups and stable under taking extension.

Proposition: every normal or quotient subgroup of a poly-simple group is poly-simple.

Proof: let $C$ be the class of groups in which every normal subgroup is poly-simple. Clearly, every simple group is in $C$. So, it's enough to show that $C$ is closed under taking extensions. Let $G$ be a group with a normal subgroup $N$, such that $N$ and $G/N$ are in $C$. Let $H$ be a normal subgroup of $G$. Then $H\cap N$ is normal in $H$, so is poly-simple, and $H/(H\cap C)$ is isomorphic to a normal subgroup of $G/N$, namely the projection of $H$ in $G/N$. Hence $H/(H\cap C)$ is poly-simple. Stability under extensions implies that $H$ is poly-simple, finshing the proof. The proof with quotients is almost the same. $\Box$

So your question has a positive answer.

Corollary: every normal subgroup of every poly-simple group lies in a Jordan-Hölder series.