**Definition 1. **Let** ** be a finite dimensional vector space over The set of all invertible -linear transformations is a group under composition of functions. This group is denoted by

**Remark 1**. We know from linear algebra that an element of corresponds to an invertible matrix where is a -basis for So if then is just the general linear group i.e. can also be viewed as the group of invertible matrices with entries in

**Definition 2**. Let be a finite group. A (complex) **representation** of is a group homomorphism The **degree** of this representation is defined to be We will denote by the degree of

**Remark 2**. Let be a representation of and Let Then where is the identity element of Thus, since is a group homomorphism, we have

where is the identity map of

**Remark 3**. A degree one representation of is just a group homomorphism where is the multiplicative group of

*Proof*. If then, as we mentioned in Remark 1, can be viewed as the group of invertible matrices with entries from which is obviously

**Remark 4**. Suppose that is a degree one representation of and let Let Then, by Remark 2 and 3, So in this case is an -th root of unity in for all