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In the book Algebra of Serge Lang, the following is written: "The fact that Hom(G,X) is a group when G,X are commutative is of special significance." (Where Hom(G,X) is the set of homomorphisms from G into X) My question is: What is the binary operation of the group Hom(G,X)?

Thank you in advance

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    No, we don't need, you are right.2012-11-06

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As Steve commented, $f+g:= \big( x\mapsto f(x)+g(x)\big)$, this is the operation (so called 'pointwise operation').

In general, for algebraic structures $(X,*)$ with one binary operation, then, the homsets $hom(X,Y)$ are naturally closed wrt. ($*$ defined pointwise) if and only if $(a*b)*(c*d)=(a*c)*(b*d)$ holds for all elements $a,b,c,d\in Y$. And, it fits perfectly for Abelian groups.