Here is an exercise from Serre's "local fiels" when he starts to do cohomology: Let G act on an abelian group A, f be an inhomogenous n cochain, i.e. $f\in C^n(G,A).$ Define an operator T on f, $Tf(g\_1,g\_2,\cdots,g\_n)=g\_1g\_2\ldots g\_n f(g\_n^{-1},g\_{n-1}^{-1},\ldots,g\_1^{-1})$. It is clear that $T^2f=f$. It is also not too hard to show $T(df)=(-1)^{n+1}d(Tf)$. Thus f is a cocycle iff Tf is, and f is a coboundary iff Tf is. When n=1, it is straightforward to see -f is cohomologous to Tf. 
Then the exercise wants us to show when n= 0,3 mod 4, f is cohomologous to Tf, 
while when n=1,2 mod 4, Tf is cohomologous to -f.  

Any idea will be appreciated.



An exercise in group cohomology
Fix the signs as Wilberd suggests in the comments, check that you get a natural automorphism of the complex computing cohomology, see that it induces in fact an automorphism of the universal $\delta$-functor $H^\bullet(G,\mathord-)$, and see what it does in degree zero.


