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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.

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    (1) is explained in more detail on the same page, under the heading "Long exact sequence of cohomology". What _precisely_ do you want to know more about?2012-05-10
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    @ZhenLin : How does the functor measures the extent in which the invariants doesn't respect exact sequences is what I am looking for. How is it done ? .2012-05-10
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    @rschwieb : Thank you for your edit.2012-05-10
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    Motivation stuff is gold: I want to make sure they don't get lost.2012-05-10
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    @Iyengar: That is measured by the long exact sequence of cohomology groups, as Wikipedia says. Do you understand what an exact sequence is?2012-05-10
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    Yes, I do understand. But I am seeing a motivation for that concept. @ZhenLin2012-05-10
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    What more is there to say? If $H^1(G; A) = 0$ then any short exact sequence $0 \to A \to B \to C \to 0$ gives a short exact sequence $0 \to A^G \to B^G \to C^G \to 0$. Otherwise you have to replace the last $0$ with $H^1(G; A)$.2012-05-10
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    Oh, Sorry sir. Thanks a lot, that explanation I was looking for. Really good. I am poor at english , and sometimes I cant understand the context and perspective. @ZhenLin2012-05-10
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    After having read the comments, I hope I haven't answered the wrong question. I interpreted you as asking where the bizarre-looking definitions of crossed homomorphisms and related objects come from. Apologies if you were looking for something else.2012-05-14
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    Can you write down what $M$ is, please, for completeness' sake?2012-05-14
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    @AlexeiAverchenko : M is an abelian group on which G acts.2012-05-19

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