A basic question of cohomology group

Question

From the material I am learning, for a chain complex $ \cdots \mathop \to \limits^\partial {C_{n + 1}}\mathop \to \limits^\partial {C_n}\mathop \to \limits^\partial {C_{n - 1}}\mathop \to \limits^\partial \cdots $, the cochain group $C_n^ * $ if defined as $C_n^ * = Hom({C_n},G)$ for a given group $G$. My question is that in what sense the cochain group $C_n^ * = Hom({C_n},G)$ is a group, should it represent a homomorphism? or $C_n^ * $ is the image of some homomorphism to $G$ and thus is a subgroup of $G$?

Answer 1

The set $Hom(C_n,G) = \{\varphi \colon C_n \to G \mid \varphi \mbox{ is a homomorphism}\}$ is a group under point-wise addition so $\varphi_1+\varphi_2$ is defined by $(\varphi_1+\varphi_2)(x) = \varphi_1(x)+\varphi_2(x)$.

This is clearly an associative and commutative binary operation (because $G$ is an abelian group).

The identity is given by the zero-homomorphism $0$ given by $0(x) = 0_G$.

The inverse $\varphi'$ of $\varphi$ is given by $\varphi'(x) = -\varphi(x)$.