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 i.e. the number of group homomorphisms is We are going to take a look at some special cases of this result.

1) If is abelian, then is trivial and hence So the number of degree one representations of a finite abelian group is just 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 is and so the number of degree one representations of must be We gave these two representations in Example 2.

3) The commutator subgroup of is Thus the number of degree one representations of is This was showed in Example 1.

4) The commuatator subgroup of the dihedral group

is Thus if is odd and if is even. Thus the number of degree one representations of is if is odd and is if is even. This result was also proved in Example 2.