# Linear character

From Groupprops

WARNING: POTENTIAL TERMINOLOGICAL CONFUSION: Please don't confuse this with Character of a linear representation

*This article is about a standard (though not very rudimentary) definition in an area related to, but not strictly part of, group theory*

This term is related to: linear representation theory

View other terms related to linear representation theory | View facts related to linear representation theory

## 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.

## Study of this notion

### Mathematical subject classification

Under the Mathematical subject classification, the study of this notion comes under the class: 20C