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\begin{document}

\maketitle

\begin{center}
{{\Large\bf ALGEBRA I}\\
\bigskip

{\Large\bf HOMEWORK ASSIGNMENT VI}\\
\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 \> 2.52, 2.54, \underline{2.55}
\end{tabbing}

}
\vspace{.5in}

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

{\bf 

1. Assume $M, N, \widehat{M}, \widehat{N}, P\lhd G$. Define $[M,N]$ to
be the subgroup of $G$ generated by $\{[m,n]\;|\;m\in M, n\in N\}$.
Show that the following are true:

\begin{enumerate}
\item[\underline{(i)}] $[M,N]\lhd G$.
\item[\underline{(ii)}] $[M,N]\subseteq M\cap N$.
\item[(iii)] $[M,N] = [N,M]$.
\item[(iv)] $[M,N\vee P] = [M,N]\vee [M,P]$.
\item[(v)] If $\widehat{M}< M$ and $\widehat{N}< N$, then 
$[\widehat{M},\widehat{N}]< [M,N]$.
\item[(vi)] $[M,N] = \{1\}$ iff the elements of 
$M$ commute with the elements of $N$.
\end{enumerate}

\bigskip

\underline{2.} Using only the properties listed in problem 1, show
that if the lattice of normal subgroups of $G$ contains
three members $M, N, P\lhd G$ that span an $M_3$:
\[M\vee N = M\vee N = N\vee P = T,\;\;\;
M\wedge N = M\wedge N = N\wedge P = B,\]
then $[T,T]\subseteq B$.

\bigskip

3. If $N\lhd G$, explain how to compute 
the commutator subgroup of $G/N$ from 
$N$ and $G' = [G,G]$.

\bigskip

4. Show that any automorphism of $S_n$ that maps some transposition 
to a transposition is an inner automorphism.
(Hints: argue that automorphisms send conjugate elements to conjugate elements,
so if an automorphism sends one transposition to another it must
send all transposition to transpositions. Then explain why any
automorphism that preserves the conjugacy class of transpositions 
must be inner.)

\bigskip

\underline{5.} If $n\neq 6$, show that $S_n$ has no outer automorphisms.
(Hint: Show that the conjugacy class of transpositions must 
be preserved, because no other conjugacy class has the same size.)

\end{document}

