Let $G$ be a group (possibly infinite). Suppose $G$ has a composition series. I could show that any other composition series has the same length. But I cannot prove the following.
Let $G \triangleright G_2 \triangleright G_3 \triangleright \cdots$ be a series of normal groups (may or may not assume the successive quotients are simple). I want to show that this series should terminate, i.e., there is $N$ such that $G_i=G_N$ for any $i >N$.
This is not true, take $G=Z^N$ and $G_i$ the subgroup such that $x\in G_i$ if its first $i$ components are zero.