Representations of finite groups; basic examples (1)

Posted: February 7, 2011 in Representations of Finite Groups
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Example 1. Let $G = \langle g \rangle$ be a cyclic group of order $n.$ Show that $G$ has exactly $n$ representations of degree one and find all of them.

Solution. If $\rho: G \longrightarrow \mathbb{C}^{\times}$ is a representation of degree one, then, as we saw in Remark 4, $\rho(g)$ is an $n$-th root of unity in $\mathbb{C}.$ Conversely, if $\zeta \in \mathbb{C}$ is an $n$-th root of unity, then the map $\rho : G \longrightarrow \mathbb{C}^{\times}$ defined by $\rho(g^j)=\zeta^j,$ for all $j,$ is obviously a well-defined group homomorphism. So, if $\zeta_1, \cdots , \zeta_n$ are the $n$-th roots of unity, then there are exactly $n$ representations of $G$ which have degree one: they are $\rho_k : G \longrightarrow \mathbb{C}^{\times}, \ 1 \leq k \leq n,$ defined by $\rho_k(g^j) = \zeta_k^{j},$ for all $j.$

Example 2. Show that the number of degree one representations of $D_{2m},$ the dihedral group of order $2m,$ is two if $m$ is odd and is four if $m$ is even.

Solution. By definition

$D_{2m}= \langle g_1,g_2: \ g_1^2=g_2^m=(g_1g_2)^2=1 \rangle.$

Every element of $D_{2m}$ is written uniquely as $g_1^j g_2^k,$ where $0 \leq j \leq 1$ and $0 \leq k \leq m-1.$ A representation of degree one for $D_{2m},$ is a group homomorphism $\rho : D_{2m} \longrightarrow \mathbb{C}^{\times}.$ Since $g_1^2=1,$ we must have $(\rho(g_1))^2=1$ and thus $\rho(g_1)=\pm 1.$ On the other hand, $\rho(g_2)$ must be an $m$-th root of unity because $g_2^m=1.$ So $\rho(g_2) = \zeta,$ where $\zeta$ is an $m$-th root of unity. So it seems that there are $m$ possibile values for $\rho(g_2)$ because there are $m$ possible values for $\zeta.$ But we also have $(g_1g_2)^2=1,$  which gives us

$1=(\rho(g_1) \rho(g_2))^2=(\pm \zeta)^2=\zeta^2.$

So $\zeta = \pm 1.$ But we also have $\zeta^m=1.$ Thus if $m$ is odd, then $\zeta=1$ and if $m$ is even, then $\zeta = \pm 1.$ So if $m$ is odd, there are only two representations of degree one for $D_{2m}$ and if $m$ is even, there are four representations of degree one for $D_{2m}.$

Example 3.  Since $S_3 \cong D_6,$ Example 2 shows that the number of degree one representations of $S_3$ is two.

Example 4. Since $K_4,$ the Klein four-group, is isomorphic to $D_4,$ Example 2 shows that the number of degree one representations of $K_4$ is four.