**Example 1**. Let be a cyclic group of order Show that has exactly representations of degree one and find all of them.

**Solution**. If is a representation of degree one, then, as we saw in Remark 4, is an -th root of unity in Conversely, if is an -th root of unity, then the map defined by for all is obviously a well-defined group homomorphism. So, if are the -th roots of unity, then there are exactly representations of which have degree one: they are defined by for all

**Example 2**. Show that the number of degree one representations of the dihedral group of order is two if is odd and is four if is even.

**Solution**. By definition

Every element of is written uniquely as where and A representation of degree one for is a group homomorphism Since we must have and thus On the other hand, must be an -th root of unity because So where is an -th root of unity. So it seems that there are possibile values for because there are possible values for But we also have which gives us

So But we also have Thus if is odd, then and if is even, then So if is odd, there are only two representations of degree one for and if is even, there are four representations of degree one for

**Example 3**. Since Example 2 shows that the number of degree one representations of is two.

**Example 4**. Since the Klein four-group, is isomorphic to Example 2 shows that the number of degree one representations of is four.