TITLE: Motivation behind the ingredients of First Cohomology group $H^1$
QUESTION [13 upvotes]: I started reading the Cohomology theory of groups. But I am not able to get any intuition or motivation behind the following :
It is concerned with the formal definitions of crossed and principal crossed homomorphisms. Crossed homomorphisms are those maps $f:G\to M$ satisfying $f(ab)=f(a)+af(b)$ ( For all $a,b \in G$ ) where as the Principal crossed homomorphisms are given by $f(a)=am-m$ for some $m \in M$. I don't really understand the motivation or the need consider the terms $f(ab)=f(a)+af(b),f(a)=am-m.$ What do they tell us ? . Do they serve as some means for calculating the so called "difference ( Given in above link ). I don't think they exist blindly or randomly. There must be some deep intuition behind that. 
I would be very happy to hear if some one posts a detailed explanation. Please name some good articles that will give a good motivation. 
Thanks a lot.

REPLY [7 votes]: Group cohomology was initially observed in nature: people noticed that the homology of certain spaces with trivial higher homotopy groups depends only on the fundamental group, and with considerable ingenuity managed to predict purely algebraically what that homology is from the fundamental group itself.
In a sense, then, people found that $H^1$ can be described in terms of crossed homomorphisms by computing the group explicitly and there is no a priori intuition of why it should be given in that way. Moreover, one can provide many equivalent descriptions of $H^1$, some of which do not involve crossed homomorphisms in any way—and, in particular, saying that crossed homomorphisms are an "ingredient" of $H^1$ is a misunderstanding: they are just one of its several avatars only...
On the other hand, people had already encountered crossed homomorphisms and principal crossed homomorphisms in nature before, independently of cohomology. I don't know who exactly made the connection, but surely someone did see that the $H^1$ that topologists found was the same one that was implicit in, say, Hilbert's Theorem 90.