Via the seminar this afternoon, we can see that one of our weak points is basic group theory, the fundamental to understand clearly about Fourier analysis on finite abelian group. We cannot go further without knowing about homomorphisms and characters. I now post some problems. SOLVE THEM ALL AND POST THE SOLUTIONS HERE! Or the seminar about Fourier analysis on finite abelian groups will STOP!
Problem 1. Construct all group homomorphisms from $\mathbb{Z}_{12}$ to $\mathbb{Z}_{18}$. In general, construct all group homomorphisms from $\mathbb{Z}_n$ to $\mathbb{Z}_m$.
Problem 2. The exponent of a finite abelian group $G$ is a minimum natural number $n$ such that $g^n=1$ for all $g\in G$. Let $G$ be a finite abelian group, and $H$ a finite cyclic group such that the exponent of $G$ divides $\#H$. Prove that $\rm{Hom}(G,H)\cong G$.
Problem 3. With $G,H$ are defined as above. Prove that for all $a\ne 1\in G$, there exists a character $\chi\in \rm{Hom}(G,H)$ such that $\chi(a)\ne 1$.
Problem 4. We define a map $\langle .,.\rangle: G\times \rm{Hom}(G,H)\rightarrow H$ that maps $\langle g,\chi \rangle$ to $\chi(g)$. Prove that
1. It is a non-degenerate pairing.
2. The set $\{\chi_g = \langle g, . \rangle|\forall g\in G\}$ is actually $\rm{Hom}(Hom(G,H),H)$.
3. The map $g\mapsto \phi_g$ is the isomorphism between $G$ and $\rm{Hom}(Hom(G,H),H)$.
Problem 5. Deduce all things to the case $H=\mathbb{C}^\times$, and write everything you know about $\mathbb{C}^G$, orthogonal basis and Fourier transform after three presentations about the topic.
You cannot say that you don't know about pairing or non-degenerate properties! Take a look on Google.
It is a very careless proof.
ReplyDeleteYou MUST write very carefully. $\chi(1)$ lies in that set. And so what? Who can ensure that $\chi(1)$ can take all values of that set? Similarly, for the second part, how can you deduce $|H| =\gcd(m,n)$? What is the group structure of the homomorphism groups? Did you construct anything? How careless you are! It is not a style of a person who considers Mathematics seriously.
ReplyDeleteProblem 2.
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I like this proof. Category theory is not only beautiful, but a powerful language, right?
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At the first glance, we can think it is a trivial problem, but actually it is not. Another proof can be done by looking at the characters of $G$ to $\mathbb{C}^\times$. If we denote $\#H=n$, then we can build a isomorphism from $H$ to $\mu_n$-the $n^\text{th}$ roots of unity in $\mathbb{C}^\times$. Via this, $Hom(G,H)\cong Hom(G,\mu_n)$, and we can write explicitly all characters of $G$ via its decomposition into cyclic groups (look at it in Tran's post about characters of finite abelian groups), and then choose $\chi$ such that $\chi(a)\ne 1$. This can give an easier proof.
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