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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?

Matt S.
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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.

DonAntonio
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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" ?? – Matt S. Nov 12 '12 at 17:27
  • 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_extension – DonAntonio Nov 12 '12 at 17:30
  • I also know what that means. The problem is with the part "built up out of the quotients factors". – Matt S. Nov 12 '12 at 18:09
  • 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? – Matt S. Nov 13 '12 at 01:32