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\title{Math 720\\Fall 2010\\Homework 1}
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\centerline{\it Due Friday September 3, 2010.}

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\noindent 1. Let $k,m,n\in\mathbb{Z}$ be nonzero integers. 
\begin{itemize}
\item[a)] (5 pt) Show that $\text{gcd}(m,n)$ is a linear combination of $m$ and $n$ (that is, show that there are integers $a$ and $b$ such that $am+bn=\text{gcd}(m,n)$).
\item[b)] (5 pt) Show that if $\text{gcd}(k,m)=1$ and $\text{gcd}(k,n)=1$, then $\text{gcd}(k,mn)=1$.
\item[c)] (5 pt) Show that if $\text{gcd}(k,m)=1$ and $k$ divides $mn$, then $k$ divides $n$.
\end{itemize}


\centerline{}

\noindent 2. ({\it 2 is, in fact, odd}) Let $G$ be a group.
\begin{itemize}
\item[a)] (5 pt) Show that any group of exponent 2 is abelian.
\item[b)] (5 pt) Show that if $G$ is finite and generated by two elements of order 2, then $G\cong D_n$ for some $n$.
\end{itemize} 
\centerline{}

\noindent 3. (5 pt) Let $S$ be a semigroup. Show that $S$ is a group if and only if $S$ be a left identity and every element of $G$ has a left inverse.

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\noindent 4. (5 pt) Show that the group $G$ is abelian if and only if the function $\phi: G\longrightarrow G$ given by $\phi(x)=x^{-1}$ is an automorphism.

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\noindent 5. (5 pt) Let $G$ be a group and $\text{Aut}(G)=\{\phi:G\longrightarrow G\vert \phi\text{ is an automorphism}\}$. If $x\in G$, we define the function $\phi_x:G\longrightarrow G$ by $\phi_x(y)=x^{-1}yx$ for all $y\in G$, and we define $\text{Inn}(G)=\{\phi_x\vert x\in G\}$. Show that $\text{Aut}(G)$ is a group with subgroup $\text{Inn}(G)$.




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