# An Exercise in Allen Hatcher's book on Spectral Sequences

anyone knows how to solve Exercise 3 of Chapter 1 of Allen Hatcher's book on Spectral Sequences? The question is as follows:

For a fibration $K(A,1)\rightarrow K(B,1)\rightarrow K(C,1)$ associated to a short exact sequence of groups $1\rightarrow A\rightarrow B\rightarrow C\rightarrow 1$ show that the associated action of $\pi_1K(C,1)=C$ on $H_*(K(A,1);G)$ is trivial if $A$, regarded as a subgroup of $B$, lies in the center of $B$.

Any help will be greatly appreciated.

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The action of $C$ on $K(A,1)$ in the case when $A$ is abelian is given by applying $K(-, 1)$ to the conjugation maps $b: A\to A$, $a\mapsto bab^{-1}$. These maps are trivial on $A$ if $A$ is abelian. Of course, all these maps are trivial if $A$ is contained in the center of $B$. I don't know how you are defining the action of $C$ on $H_*(A, M)$ so I don't know how to prove these are the same. –  Justin Young Apr 18 '12 at 18:54