**Example 1**. Find all degree one representations of the quaternion group

**Solution**. We will show that there are exactly four degree one representations of We know that

There is a another way to define if we use relations:

Note that and in (2) are just and in (1). Anyway, suppose that is a representation of Then because If then since we will get which is absurd. Thus and so, since we get and So there are four possible ways to define Note that either of these four ways is consistent with the relations in (2) because

**Example 2**. In this example we will give two degree one representations of the group of permutations of We will see later that these two representations are actually the only degree one representations of The first representation is a trivial one: define by for all The other representation is called which maps every to the signature of Recall that the signature of a permutation is defined to be if is even and if is odd. We know from group theory that for all So is a representation.