Abstract Nonsense

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.

Motivation

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 $G$ is cyclic then $G\cong\widehat{G_\mathfrak{L}}$.

Dual Group of a Cyclic Group

We now proceed to show that if $G$ is cyclic then $G\cong \widehat{G_\mathfrak{L}}$. The key fact is that it will suffice to prove the result for $\mathbb{Z}_n$ for a fixed but arbitrary $n$ and really to show that $\text{Hom}\left(\mathbb{Z}_n,\mathbb{T}\right)\cong\mathbb{Z}_n$ from where the conclusion follow from previous theorem.

Theorem: Let $G$ be a finite cyclic group, then $G\cong\widehat{G_\mathfrak{L}}$.

Proof: As previously stated if we prove that $\text{Hom}\left(\mathbb{Z}_n,\mathbb{T}\right)\cong U_n$ where $U_n$ is the group of $n^{\text{th}}$-roots of unity and $n=|G|$ we’ll be done since

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$G\cong\mathbb{Z}_n\cong U_n\cong\text{Hom}\left(\mathbb{Z}_n,\mathbb{T}\right)\cong\text{Hom}\left(G,\mathbb{T}\right)\cong\widehat{G_\mathfrak{L}}$

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So, define $F:\text{Hom}\left(\mathbb{Z}_n,\mathbb{T}\right)\to U_n$ by $\chi\mapsto \chi(1)$. This is evidently a homomorphism since $F(\chi\lambda)=(\chi\lambda)(1)=\chi(1)\lambda(1)=F(\chi)F(\lambda)$. It’s injective since $1$ is a generator for $\mathbb{Z}_n$ and thus by basic group theory if $F(\chi)=\chi(1)=\lambda(1)=F(\lambda)$ then $\chi=\lambda$. Lastly, to see it’s surjective we merely note that for any $\zeta\in U_n$ defining $\chi:\mathbb{Z}_n\to U_n$ by $\chi(k)=\zeta^k$ is a homomorphism and thus $\chi\in\text{Hom}\left(\mathbb{Z}_n,\mathbb{T}\right)$. But, this then implies that $\zeta=\chi(1)=F(\chi)\in\text{im}(F)$ from where surjectivity follows. It follows then that $F$ is an isomorphism and thus the entire theorem follows from previous discussion. $\blacksquare$

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References:

1. Simon, Barry. Representations of Finite and Compact Groups. Providence, RI: American Math. Soc., 1996. Print.

April 13, 2011 -