In this post we are going to look at degree two representations of the dihedral group of order We will construct such representations.

**Example 1**. Let be the dihedral group of order as described in Example 2. Then, as we mentioned in that example, every element of is written uniquely as where and Let

Let be an -th root of unity and define by

and

for all Clearly and are isomorphisms and so they are in It is easily seen that

That means and satisfy the same relations as do. So the map defined by

for and is a well-defined group homomorphism. Thus is a representation of and This gives us representations of degree two for because the number of -th roots of unity is

**Remark**. As I mentioned in Remark 1, the elements of may also be viewed as invertible matrices: let be an -th root of unity and consider with the standard basis. Let

and

The linear transformation corresponding to (resp. ) is clearly (resp. ) defined in the above example. It might be easier sometimes to work with corresponding matrices instead of linear transformations. So the matrices corresponding to the representations defined in Example 1 are

and

where is an -th root of unity and

**Example 2**. Since Example 1 gives us three degree two representations of

**Example 3**. Since the Klein four-group is isomorphic to Example 1 gives us two degree two representations of