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In the context of normal and subnormal series I've found the following:

"From a finite subnormal series of a group $G$ we obtain a sequence of exact sequences and thus $G$ is built up out of the quotients factors of the sequence by forming successive extensions."

Which is the formalism to express this group $G$ by this way?

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$N\triangleleft G\Longrightarrow\,\text{we get the exact sequence}\,\, 1\longrightarrow N\stackrel{i}\longrightarrow G\stackrel{\pi}\longrightarrow G/N\longrightarrow 1$

with $\,i=\,$ the embedding injection and $\,\pi=\,$ the natural (surjective, of course) projection.

Now do the same as above for any subnormal series and every two elements in it.

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    You said "Now do the same as above for any subnormal series and every two elements in it".But this is the exactly point which I don't know what to do next. What should I do after that?2012-11-13