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I have been studying the concept of Schur multiplier $M(G)$ of a finite group $G$, and their relation with possible central extensions of $G$.

I read that if $G$ is a perfect group ($G=[G,G]$) then there exists a (unique) universal central extension $\tilde{G}$ of $G$, which has $M(G)$ as kernel of the $\tilde{G} \to G$ map. It being universal means that for any other central extension $H$ there exists a homomorphism $\tilde{G} \to H$ compatible with the extension maps.
However, I do not understand in what sense does this "universal property" help classifying them, since (for example) for any abelian group $A$, $G \times A$ is a central extension of $G$ (so possible central extensions are not bounded in size).

If the group $G$ is not perfect, I haven't been able to find much.

Hence, my question: in what way, and up to what, does knowing the Schur multiplier of a group help us classify the possible central extensions? In particular, what can we say for a group $G$ with trivial Schur multiplier, and its non-split extensions? If a full answer is too long or elaborate, I also appreciate references.

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  • $\begingroup$ I don't think I could do a full answer justice, but Finite Group Theory by Micheal Aschbacher has a very nice section on central extensions which I think answers your question nicely $\endgroup$ – Robert Chamberlain May 18 '17 at 16:25
  • $\begingroup$ Oh, and this is where I look stupid :( Before asking this question I was actually reading Aschbacher's book, and I understand the single propositions, but I haven't been able to get a clear picture of what we can actually say, given a group $G$ and its Schur multiplier $M(G)$. Being more precise, it seems to me that we can't actually say much if $G$ is not perfect... $\endgroup$ – AnalysisStudent0414 May 18 '17 at 16:56
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    $\begingroup$ For example, for the group $A_5 \cong {\rm PSL}(2,5)$, the universal covering group is the $2$-fold central extension ${\rm SL}(2,5)$. So an arbitrary central extension of $A_5$ is either a direct product $A_5 \times A$ or ${\rm SL}(2,5) \times A$, for an arbitrary abelian group $A$, or else it could be a central product of ${\rm SL}(2,5)$ with an abelian group that contains an element of order $2$. That is what I would call a classification! $\endgroup$ – Derek Holt May 18 '17 at 20:02
  • $\begingroup$ By the way, I don't why you have assumed that $G$ is finite. None of what you have written requires $G$ to be finite. $\endgroup$ – Derek Holt May 18 '17 at 20:04
  • $\begingroup$ I am aware of that! I was hoping for the finiteness of $G$ to somehow imply stronger results... thank you for your comment! $\endgroup$ – AnalysisStudent0414 May 18 '17 at 21:34

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