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\begin{center}
  \textbf{Math 71}\\ Homework 5
\end{center}




\begin{enumerate}
\item As usual, for positive integers, let $[m,n]$ denote the least
  common multiple of $m$ and $n$, and $(m,n)$ their gcd.

  \begin{enumerate}
  \item \textbf{Using the First Isomorphism Theorem} show that there
    is an injective homomorphism $\Z/[m,n]\Z \to \Z/m\Z \times
    \Z/n\Z$.  Note: to use the first isomorphism theorem means you do
    \textbf{not} (directly) define a homomorphism $\Z/[m,n]\Z \to
    \Z/m\Z \times \Z/n\Z$.

  \item Show that the above mapping is surjective (and hence an
    isomorphism) if and only if $(m,n) = 1$, in which case $\Z/mn\Z
    \cong \Z/m\Z \times \Z/n\Z$.
  \end{enumerate}
\item Let $G_1 = \la x \ra$ and $G_2 = \la y \ra$ be finite cyclic
  groups of order $m$ and $n$ respectively.  Show that $G_1 \times
  G_2$ is cyclic if and only if $(m,n) = 1$. This should follow easily
  from the previous problem.

\item Let $\varphi: \Z/18\Z \to \Z/12\Z$ be the homomorphism
  satisfying $\varphi( \bar 1 ) = \bar{10}$, that is the class of 1 in
  $\Z/18\Z$ is mapped to the class of $10$ in $\Z/12\Z$.
  \begin{enumerate}
  \item Show that $\varphi$ is well-defined, but the map $\psi:
    \Z/18\Z \to \Z/12\Z$ induced by $\psi(\bar 1) = \bar 3$ is not.

  \item Compute the kernel of $\varphi$, and apply the first
    isomorphism theorem.

  \item To what well-known group is $Im(\varphi)$ isomorphic?  Using
    this representation, your statement invoking the first isomorphism
    theorem should look like an application of the third isomorphism
    theorem; make sure it does!
  \end{enumerate}

\item Let $\Z[x]$ denote the set of polynomials with coefficients in
  $\Z$.  It is obvious that $\Z[x]$ is an additive group .  The
  example will be even more important as we start the study of rings.
  \begin{enumerate}
  \item Define $\varphi: \Z[x] \to \Z$ by $\varphi(f) = f(0)$.  Show
    that $\varphi$ is a surjective homomorphism and compute its
    kernel.  What statement does the first isomorphism theorem yield?

  \item Do the same as above with $\varphi: \Z[x] \to \Z$ defined by
    $\varphi(f) = f(3)$.
  \end{enumerate}


\item page 122: 8, 10, 14


\end{enumerate}




 

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