# Showing that a subnormal series for a finite group $G$ can be made into a composition series for $G$

Suppose that $\{e\} < G_1 < G_2 < \cdots < G_n = G$ is a subnormal series, thus for all $i$, we have $G_i$ is normal in $G_{i+1}$. How can I show that this can be "refined" to a composition series?

By composition series I mean a subnormal series where, for each $i, G_{i+1}/G_i$ is a nontrivial, simple group.

• I doubt it always can. For example, without restrictions on the group, such a refinement may require an infinite number of factors. – Mose Wintner May 30 '16 at 20:27
• For example it is not possible for the series $\{e \} < {\mathbb Z}$ of ${\mathbb Z}$. The result is clear for finite groups. If it is not composition series already then some factor $G_{i+1}/G_i$ is not simple, so it has a proper nontrivial normal subgroup $N/G_i$. So just insert $N$ into the series between $G_i$ and $G_{i+1}$. Keep doing this until you get a composition series. – Derek Holt May 30 '16 at 20:29
• @Mose Winter, that's true, I just realized I left something important out. We have to assume the group is finite. – Ninosław Brzostowiecki May 30 '16 at 20:29

Suppose $G_{n+1}/G_n$ is not simple, then we have $NG_n$ an intermediate normal subgroup. Rinse and repeat.