Constructing a Normal Series for G from Given Normal Series for N and G/N

[Chaos2009]

Homework Statement

Suppose that $N \triangleleft G$. Show that given normal series $S$ for $N$ and $T$ for $G / N$ one can construct a normal series $U$ for $G$ such that the first part of $U$ is isomorphic to $S$ and the rest is isomorphic to $T$.

Homework Equations

This is from the last couple of weeks of an undergraduate Abstract Algebra course. The teacher assigned it as homework while discussing a proof of the Jordan-Holder theorem.

The Attempt at a Solution

I'd like to simply construct $U$ from $S$ and $T$. Using $S$ would be straightforward as this is already a normal series from $\left\{ e \right\}$ to $N$. However, I'd hoped to use correspondence theorem to map the normal series $T$ to a normal series from $N$ to $G$. I believe, however that there is a problem with the part where it says this part of the series should be isomorphic to $T$.

 

[morphism]

What does it mean to say that two normal series are isomorphic?

 

[Chaos2009]

I guess that was the part I was confused about as well. My roommate now informs me that we defined two normal series to be isomorphic as follows:

Series $S$ and $T$ are isomorphic if there exists a bijection from the factors of $S$ to the factors of $T$ such that the corresponding factors are isomorphic.

So, that makes a bit more sense to me now.

 

[morphism]

Typically that's referred to as "equivalence", but anyway, your idea does work, i.e. it will produce an equivalent normal series.