(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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

2. Relevant 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.

3. The attempt at a solution

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

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# Homework Help: Proving Jordan-Holder theorem

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