The Dual Group of a Cyclic Group
Point of post: In this post we show that the dual group of cyclic groups are very simple, in fact they are isomorphic to the original group itself.
Often it’s difficult to deal directly with the dual group even with our alternate characterization of it. In this post we show that the lay of the land becomes much nicer. Namely, we shall show that if is cyclic then .
Dual Group of a Cyclic Group
We now proceed to show that if is cyclic then . The key fact is that it will suffice to prove the result for for a fixed but arbitrary and really to show that from where the conclusion follow from previous theorem.
Theorem: Let be a finite cyclic group, then .
Proof: As previously stated if we prove that where is the group of -roots of unity and we’ll be done since
So, define by . This is evidently a homomorphism since . It’s injective since is a generator for and thus by basic group theory if then . Lastly, to see it’s surjective we merely note that for any defining by is a homomorphism and thus . But, this then implies that from where surjectivity follows. It follows then that is an isomorphism and thus the entire theorem follows from previous discussion.
1. Simon, Barry. Representations of Finite and Compact Groups. Providence, RI: American Math. Soc., 1996. Print.