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.