Linear character

Definition
A linear character of a group over a field is a homomorphism from the group to the multiplicative group of the field. Equivalently, it can be thought of as a representation of the group over the field, whose degree (dimension) is one.

People working with Abelian groups typically use the term character to refer to a linear character over the complex numbers that always take values on the unit circle (in other words, a unitary linear character). For them, a character is thus simply a homomorphism to the circle group.