I'm having trouble understanding how to show that the product of a homomorphism into an abelian group H, from a group G is a homomorphism. First, the operation would be $fb(g)$=$f(g)b(g)$. I am wondering how to show that this operation is associative. Thanks!
Edit: I figured out why it is a homomorphism! Could anyone just help me show that the product of homomorphisms from G to an abelian group H is associative?
The group operations (on $G$ and $H$) are given and are assumed to be associative.
The definition of a homomorphism is that the given function has to respect the respective group operations: $$u(gh)=u(g)u(h)\quad\ u(e_G)=e_H$$ You only have to verify these for the product $u=fb$ defined above, using commutativity in $H$.