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\begin{center}\begin{large}\textbf{Math 101 - Problem Set 8\\
Due Tuesday, November 7}\end{large}\end{center}


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\begin{enumerate} 

\item Consider the following groups of order 8: $D_8, Q_8, \Z_{8}, \Z_4 \times \Z_2.$
For each pair of groups, either show that the two groups are isomorphic, or explain why they're not.

\item Determine whether each of the following functions is a homomorphism or not. 

\begin{enumerate}


\item $\phi: \R -\{0\} \to \R -\{0\}$ under multiplication, defined $\phi(x) =$ the absolute value of $x$.

\item $\phi: \R \to \Z$ under addition, defined $\phi(x) =$ the greatest integer less than or equal to $x$ (For example, $\phi(3.7) = 3$).

\item $\phi: \Z \times \Z \to \Z$ under addition, defined $\phi(x,y) = x-y$.

\item $\phi: D_8 \to \Z_4$ defined $\phi(r^i s^j ) = i$ (mod 4). 
\end{enumerate}

\item For each of the functions in the previous problem that are in fact homomorphisms, describe the kernel and the image. 

\item Let $G$ be a group. Show that the function $f:G \to G$ defined $f(x) = x^{-1}$ is a homomorphism if and only if $G$ is abelian.

\item Show that $\Z \times \Z$ (under addition) is not cyclic.

\item Show that every subgroup of a cyclic group is cyclic.

\item Using the statement you proved in the previous problem, list each subgroup of $\Z_{8}$. How many distinct subgroups are there? How many elements of $\Z_8$ generate the entire group?




\end{enumerate}
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