I am having trouble with the following problem, taken from Herstein's Topics in Algebra 2nd Edition:

If $G$ is an abelian group, let $\hat G$ be the set of all homomorphisms of $G$ into the group of nonzero complex numbers under multiplication. If, $\phi_1,\phi_2\in \hat G$, define $\phi_1 \cdot \phi_2$ by $\phi_1 \cdot \phi_2(g) = \phi_1(g) \phi_2 (g)$ for all $g \in G.$

Show that, given $G$ is also finite and cyclic, $\hat G$ is cyclic and $G$ and $\hat G$ are isomorphic.

I have already proved that $\hat G$ is abelian and that for $\phi \in \hat G$ we have $\phi(g)$ a root of unity for every $g \in G$.

My approach:

Since $G$ and $\hat G$ are both finite and abelian, we have $G=A_1 \times ...\times A_n$ and $\hat G=B_1 \times ... \times B_m$ where each $A_i$ and $B_i$ are cyclic.

Now I want to show that $n=m$ and that $A_i=B_i$ for each $i=1,..n$.

Is this a proper approach? If so, any suggestions on how I can go about this?

  • $\begingroup$ Isn't the problem assuming that $G$ has only one cyclic factor? $\endgroup$
    – Will Dana
    Nov 16 '16 at 0:38
  • $\begingroup$ It will be easier to show your groups have the same order, and both are cyclic. $\endgroup$
    – Steve D
    Nov 16 '16 at 0:44
  • $\begingroup$ @WillDana oh right, of course, and that immediately shows that $\hat G$ is cyclic, since it is abelian and each of its elements is a root of unity. $\endgroup$
    – user389773
    Nov 16 '16 at 0:45
  • $\begingroup$ @SteveD How should I start to prove they have the same order? $\endgroup$
    – user389773
    Nov 16 '16 at 0:48
  • $\begingroup$ Every map is defined by where it sends a generator of $G$. How many complex numbers can be the image of such a generator? $\endgroup$
    – Steve D
    Nov 16 '16 at 2:26

First note that for a finite cyclic group $G$ any homomophism to any other group $H$ is known if the image of a generator $g$ of $G$ is known since the images of $g^2,g^3, \ldots$ can be calculated. As you stated correctly the images in this case are all roots of unity. If these are taken as the image of a certain homomophism what have these elements in common (w.r.t. their order)? What can you say about an arbitrary homomorphism w.r.t. a fixed homomorphism that maps $g$ to a primitive root of unity?


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