## Number of representations of degree one (2)

Posted: February 8, 2011 in Representations of Finite Groups
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In this post I’ll be looking at the theorem we proved in this post in connection with previous examples. In that theorem we peoved that the number of degree one representations of a finite group $G,$ i.e. the number of group homomorphisms $\rho : G \longrightarrow \mathbb{C}^{\times},$ is $[G:G'].$ We are going to take a look at some special cases of this result.

1) If $G$ is abelian, then $G'$ is trivial and hence $[G:G']=|G|.$ So the number of degree one representations of a finite abelian group $G$ is just $|G|.$ This is the lemma we proved in this post. Also, since every cyclic group is abelian, we will recover the result in Example 1.

2) The commutator subgroup of $S_n$ is $A_n$ and so the number of degree one representations of $S_n$ must be $[S_n:A_n]=2.$ We gave these two representations in Example 2.

3) The commutator subgroup of $Q_8$ is $\{1,-1\}.$ Thus the number of degree one representations of $Q_8$ is $[Q_8 : \{1,-1\}]=4.$ This was showed in Example 1.

4) The commuatator subgroup of the dihedral group

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

is $\langle g_2^2 \rangle.$ Thus $|D_{2m}'|=m$ if $m$ is odd and $|D_{2m}'|=m/2$ if $m$ is even. Thus $[D_{2m}: D_{2m}'],$ the number of degree one representations of $D_{2m},$ is $2$ if $m$ is odd and is $4$ if $m$ is even. This result was also proved in Example 2.