Representations of finite groups; basic examples (1)

Posted: February 7, 2011 in Representations of Finite Groups
Tags: , , ,

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.

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