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for a finite group G and a trivial abelian G module A, there is the short exact sequence

$0 \rightarrow Ext^1 (G_{ab},A) \rightarrow H^2(G,A) \rightarrow Hom (H_2 (G,Z), A) \rightarrow 0$

I'm looking for a description for the connecting maps. Specially, I want to use the representation of $H_2(G,Z)$ as $M(G)=[F,F]\cap R/ [F,R]$ where $G\cong F/R$ and F is free.

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up vote 3 down vote accepted

By «connecting maps» you mean the maps that appear in the short exact sequence you wrote?

In any case, you will find the details of the construction of the sequence in Hilton-Stammbach's book A Course on homological algebra, GTM4. The result is theorem VI.15.1, but the meat of the construction is in section §V.3.

If you want the expressions in terms of the isomorphisms between $H_2(G,\mathbb Z)$ and $M(G)$, well, then you have to write down both and compose them :) I do not have time now to do the computation, but you should really do it yourself!

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