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\title{\textbf{Homework 8}}

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\section*{Section 9.4}

\textbf{Exercise 2.}
\vskip1ex
Prove that $\C^*$ is isomorphic to the subgroup of $GL_2(\R)$ consisting of matrices of the form $\Big ( \begin{matrix} a & b \\ -b & a \end{matrix} \Big )$.
\vskip1ex
\emph{Proof: }
\vskip1ex

\hfill $\square$

\vskip2ex

\textbf{Exercise 7.}
\vskip1ex
Show that any cyclic group of order $n$ is isomorphic to $\Z_n$.
\vskip1ex
\emph{Proof: }

\hfill $\square$

\vskip2ex

\textbf{Exercise 8.}
\vskip1ex
Prove that $\Q$ is not isomorphic to $\Z$.
\vskip1ex
\emph{Proof: }

\hfill $\square$

\textbf{Exercise 9.}
\vskip1ex
Let $G = \R \setminus \{-1 \}$ be a group with the binary operation $a \ast b = a+b+ab.$ (see the proof in HW3.)
Show that $G \cong \R^*$.
\vskip1ex
\emph{Proof: }

\hfill $\square$

\vskip2ex

\textbf{Exercise 15.}
\vskip1ex
List all of the elements of $\Z_4 \times \Z_2$.
\vskip1ex
\emph{Solution: }

\hfill $\square$

\vskip2ex

\textbf{Exercise 16 (b and d).}
\begin{enumerate}
\item[(b)] Find the order of $(6,15,4)$ $\Z_{30} \times \Z_{45} \times \Z_{24}$. 
\vskip1ex
\emph{Solution: }
\vskip1ex
\item[(d)] Find the order of $(8,8,8)$ in $\Z_{10} \times \Z_{24} \times \Z_{80}$.
\vskip1ex
\emph{Solution: }
\end{enumerate}

\hfill $\square$

\vskip2ex

\textbf{Exercise 24.}
\vskip1ex
Prove or disprove: there is a noncyclic abelian group of order $51$.
\vskip1ex
\emph{Solution: }

\hfill $\square$

\vskip2ex

\textbf{Exercise 25.}
\vskip1ex
Prove or disprove: there is a noncyclic abelian group of order $52$.
\vskip1ex
\emph{Solution: }

\hfill $\square$

\vskip2ex

\textbf{Exercise 34.}
\vskip1ex
Show that the map $\phi(a+bi) = a-bi$ is an isomorphism from $\C$ to $\C$, i.e. an automorphism. 
\vskip1ex
\emph{Proof: }

\hfill $\square$

\vskip2ex

\textbf{Exercise 41.}
\vskip1ex
Let $G$ be a group and $g \in G$. Define $i_g: G \to G$ by $i_g(x) = gxg^{-1}$. 
Prove that $i_g$ defines an automorphism of $G$ (called an inner automorphism, Inn($G$)).
\vskip1ex
\emph{Proof: }

\hfill $\square$

\vskip2ex

\textbf{Exercise 46.}
\vskip1ex
Prove that if $G \cong H $ then if $G$ has a subgroup of order $n$, $H$ must also have a subgroup of order $n$.
\vskip1ex
\emph{Proof:}

\hfill $\square$

\vskip2ex

\textbf{Exercise 50.} Prove that $A \times B$ is abelian if and only of $A$ and $B$ are abelian.
\vskip1ex
\emph{Proof: }

\hfill $\square$

\end{document} 
