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.