## Representations of finite groups; basic examples (2)

Posted: February 8, 2011 in Representations of Finite Groups
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Example 1. Find all degree one representations of the quaternion group $Q_8.$

Solution. We will show that there are exactly four degree one representations of $Q_8.$ We know that

$Q_8 = \{ \pm 1, \pm i, \pm j, \pm k \}. \ \ \ \ \ \ \ \ \ \ \ (1)$

There is a another way to define $Q_8$ if we use relations:

$Q_8 = \langle a,b : \ a^4=b^4=1, \ a^2=b^2, \ aba=b \rangle. \ \ \ \ \ \ \ \ \ \ (2)$

Note that $a$ and $b$ in (2) are just $i$ and $j$ in (1). Anyway, suppose that $\rho : Q_8 \longrightarrow \mathbb{C}^{\times}$ is a representation of $Q_8.$ Then $\rho(-1) = \pm 1$ because $(\rho(-1))^2=\rho(1)=1.$ If $\rho(-1)=-1,$ then since $ij=-ji,$ we will get $\rho(i) \rho(j) = -\rho(i) \rho(j),$ which is absurd. Thus $\rho(-1)=1$ and so, since $(\rho(i))^2=(\rho(j))^2=\rho(-1)=1,$ we get $\rho(i) = \pm 1$ and $\rho(j)=\pm 1.$ So there are four possible ways to define $\rho.$ Note that either of these four ways is consistent with the relations in (2) because $\rho(-1)=1.$

Example 2.  In this example we will give two degree one representations of $S_n,$ the group of permutations of $\{1,2, \cdots , n \}.$ We will see later that these two representations are actually the only degree one representations of $S_n.$ The first representation is a trivial one: define $\rho : S_n \longrightarrow \mathbb{C}^{\times}$ by $\rho(\sigma)=1,$ for all $\sigma \in S_n.$ The other representation is called $\text{sgn},$ which maps every $\sigma \in S_n$ to $\text{sgn}(\sigma),$ the signature of $\sigma.$ Recall that the signature of a permutation $\sigma \in S_n$ is defined to be $1$ if $\sigma$ is even and $-1$ if $\sigma$ is odd. We know from group theory that $\text{sgn}(\sigma_1 \sigma_2)= \text{sgn}(\sigma_1) \text{sgn}(\sigma_2),$ for all $\sigma_1, \sigma_2.$ So $\text{sgn} : S_n \longrightarrow \mathbb{C}^{\times}$ is a representation.