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\begin{center}
{{\Large\bf MODERN ALGEBRA II}\\
\bigskip

{\Large\bf HOMEWORK ASSIGNMENT II}\\
\bigskip
{\bf (Turn in underlined problems.)}}
\end{center}

\vspace{.5in}

\noindent
{\bf 
\begin{tabbing}
\underline{\large\bf CHAPTER}\makebox[2in]{}\=\underline{\large\bf PROBLEMS}\\

\bigskip
\noindent
2 \> \underline{2.11}, 2.12,
\underline{2.19},
\underline{2.20}
\end{tabbing}

}
\vspace{.5in}

\noindent
\underline{\large\bf ADDITIONAL PROBLEMS}
\vspace{.25in}

{\bf 

\underline{1.}
Let {\bf G} be a group, and let $g\in G$ be any element.
Show that the function $\gamma_g:{\bf G}\rightarrow {\bf G}$
defined by $\gamma_g(x) = gxg^{-1}$ is an automorphism
of {\bf G}. (Automorphisms of this type are called
{\it inner automorphisms}. An automorphism which is
not an inner automorphism is an {\it outer automorphism}.)

\bigskip

2. Show also that the function 
$\gamma:{\bf G}\rightarrow {\rm Aut}({\bf G})$ 
defined by $\gamma(g) = \gamma_g$ is 
a homomorphism.

\bigskip

\underline{3.}
Describe all automorphisms of the group ${\BZ}_6$.

\bigskip

For Graduate Credit:

B. Do problem {2.21}.

}
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