# short exact sequence - split, as a semidirect product, with some cohomology

I've looked at several s.e.s. examples and I feel I am quite close but here is a question I am still a little confused on.

Let $E$ be a group and $A$ an abelian normal subgroup s.t. have an exact sequence, $$1 \to A \overset{i}{\to} E \overset{p}{\to} G \to 1,$$ and so $G \cong E/A$ (extension). Then let $s: G \to E$ be any choice of coset representative (set-theoretic map) s.t. $p \circ s = id_G$. If $g,\,g' \in G$, define $$f(g,g'):=s(g)\,s(g')\,s(g\,g')^{-1}.$$

1.)Show that $f$ is $A$-valued and that $f: G\times G \to A$ is a factor system, i.e., that it belongs to $ker\,(d_2)$. Action of $G$ on $A$ is defined as $$g\circ a = s(g)\,a\,s(g)^{-1}.$$

2.) Check that this action is well-defined, i.e., independent of choice of $s$.

3.) Show that if $s': G \to E$ is another choice of coset representative, then the corresponding map $f'$ and $f$ differ by an element in $Im\,(d_1)$.

So here is what I have so far. I understand it to be the chain given above, with $G \cong E/A$, and the map $s: G \to E$ splitting (epi style) the second `half'. I like to think of $s': E/A \to G$ (is this okay?)

For 1.) - if I knew what form the coboundary map(?) was for the cohomology, like $d_n(f)()$ I would crank out $d_2$ and look at the homogeneous form and see what satisfies being sent to the identity map. But I don't have that. I'm not even sure, but I think there would be a cohomology by $$0 \to E_0(A \to E) \to E_1(E \to G) \to E_2(G \to 1) \to 0$$ (sorry for the probably heinous abuse of notation) My guess is that this $f(g,g')$ is somehow related to $d_3$. ?

2.) As I understand the zeroth cohomology group, in some situations, the homogeneous form of the kernel, $d_{0}$, shows that a subgroup of some group is invariant under the group action, e.g., like $d_0(f)(g) = g\,a -a$. Similarily, though not as well understood by me, is how the trivial crossed homomorphism mods out the principal crossed homomorphism. So I expect any element in a coset that is invariant to the group action to remain invariant - if I am right how do I show $s$ doesn't matter.

3.) Lastly, and closely related to 2.), is I think that a coset rep from a different coset will .. well I'm not sure on this one.

Thank you in advance for any suggestions!

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I'm not sure this question has a very good answer. I'm not sure why one would care if the factor set is in the kernel or image of $d_i$ is one is not sure what $d_i$ is. #2 has nothing to do with cohomology. If you are trying to learn about extensions, forget about $d_i$ and classify the $f$ from #1 and the differences from #3; you'll discover factor sets and crossed homomorphisms and understand why they are important. If you are trying to learn about $d_i$, then you'll want to know their definition, but definitions are better found in books or lecture notes than on a Q-and-A site. –  Jack Schmidt Oct 2 '12 at 22:10
@Jack I appreciate your feedback - Serge Lang's book, Algebra - Chapter XX on Cohomology, question 5 - I'm just trying to figure out the way to answer it that is contextually relevant to that chapter. –  nate Oct 2 '12 at 22:22