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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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    Ok. I did. This is the first part of the claim. What about the part "G is built up out of the quotients factors of the sequence by forming successive extensions" ??2012-11-12
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    Do you kinow what does "The group $\,G\,$ is an extension of some (normal) subgroup $\,N\,$ by a group $\,K\,$" mean? Take a peek at this site:http://en.wikipedia.org/wiki/Group_extension2012-11-12
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    I also know what that means. The problem is with the part "built up out of the quotients factors".2012-11-12
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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