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\pagestyle{headandfoot} \header{\bf Abstract Algebra - Homework 10}{}{ Fall 2014} \footer{}{\thepage}{}
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\begin{enumerate}
\item (+) Let $G=\mathbb{R}[x]$ denote the group of all polynomials with real coefficients under addition.\\  (a) For any $f\in \mathbb{R}[x]$, define $\phi: \mathbb{R}[x]\ra \mathbb{R}[x]$ by $\phi(f) =f'$, the derivative of $f$.  Show that $\phi$ defines a homomorphism. \\
(b) Let $\int f$ denote the antiderivative of $f$ through the point $(0,0)$.  Show that the mapping $f\mapsto \int f$ from $G$ to $G$ defines a homomorphism.  What is the kernel of this mapping?\\
(c) Is the map in (b) still a homomorphism if $\int f $ denotes the antiderivative of $f$ through the point $(0,1)$?

 \item (+) Prove that the mapping $\phi: \mathbb{Z}\times \mathbb{Z} \rightarrow \mathbb{Z}$ given by $(a,b)\mapsto a-b$ is a homomorphism. What is the kernel of $\phi$?  Describe the set $\phi^{-1}(3)$ (that is, all elements that map to 3).

\item In the following, compute the indicated quantities for the given homomorphism. \begin{enumerate}
\item $\Ker(\phi)$ and $\phi(18)$ for $\phi: \Z \ra S_{8}$ such that $\phi(1)=(1426)(257)$. 
\item $\Ker(\phi)$ and $\phi(-3,2)$ for $\phi: \Z\times \Z \ra \Z$ where $\phi(1,0)=3$ and $\phi(0,1)=-4$
\item (+) $\Ker(\phi)$ and $\phi(15)$ for $\phi:\Z_{24} \ra S_{8}$ where $\phi(1) =(25)(1467)$.  

\end{enumerate}

\item Suppose that $k$ is a divisor of $n$.  Prove that $\Z_n/\langle k \rangle \simeq \Z_{k}$.

\item (+) Suppose that $\phi: \Z_{40}^* \ra \Z_{40}^*$ is a homomorphism.  (Recall, $\Z_{40}^*$ is the group of all elements of $\Z_{40}$ that have a multiplicative inverse with operation $\cdot$ modulo $40$.)  If $\Ker\phi = \{1,9, 7,17,33\}$ and $\phi(11)=11$, find all elements of $\Z_{40}^*$ that map to $11$.

\item For the following, give an example of a nontrivial homomorphism $\phi$ for the given groups, if an example exists.  If no such homomorphism exists, explain why that is so. \begin{multicols}{3} \begin{enumerate} \item (+) $\phi: \Z_{12} \ra \Z_{5}$  \item $\phi: \Z \ra S_{3}$  \item $\phi: \Z_{12} \ra \Z_{4}$ \end{enumerate} \end{multicols}

\item Let $\phi: G \ra G'$ be a homomorphism with kernel $H$ and let $a\in G$.  Prove the set equality $\{x\in G \mid \phi(x) = \phi(a)\}=Ha$.

\item (*) Let $G$ be a group, $h\in G$ and $n\in \Z^{+}$.  Let $\phi: \Z_{n} \ra G$ be defined by $\phi(i) = h^i$ for $0\leq i \leq n$.  Give a necessary and sufficient condition (in terms of $h$ and $n$) for $\phi$ to be a homomorphism.  Prove your assertion. 
 

\item (+) Let $G$ be a group that is generated by the set $\{a_{i} \mid i\in I\}$ where $I$ is some indexing set and $a_{i} \in G$ for all $i\in I$.  Let $\phi: G\ra G'$  and $\mu: G \ra G'$ be two homomorphisms from $G$ into a group $G'$ such that $\phi(a_i) = \mu(a_i)$ for all $i\in I$.  Prove that $\phi=\mu$. (This shows that any homomorphism of $G$ is determined by its action on the generators of $G$.)  

\item How many homomorphisms are there of $\Z$ into $\Z_2$?




\end{enumerate}

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