A character on a group $G$ is a homomorphism from $G$ into the multiplicative group of a field.

One way that characters arise is via representation theory. Given a representation of a group $\rho : G \to \operatorname{GL}(V)$ where $V$ is a finite-dimensional vector space over the field $\mathbb{F}$, the homomorphism $\chi : G \to \mathbb{F}^{\times}$ given by $\chi(g) = \operatorname{trace}(\rho(g))$ is called the character of the representation $\rho$.

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