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\title{\textbf{Homework 6}}

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\section*{Section 4.5}
\vskip2ex
\textbf{Exercise 6.} \vskip1ex
Find the order of every element in the symmetry group of the square, $D_4$.
\vskip1ex
\emph{Solution:}
\hfill $\square$

\vskip2ex

\section*{Section 5.4}
\vskip2ex
\textbf{Exercise 1.} \vskip1ex
Write the following permutations in cycle notation.
\begin{enumerate}
\item[(a)]
$\begin{pmatrix}
1 & 2 & 3 & 4 & 5 \\
2 & 4 & 1 & 5 & 3
\end{pmatrix}$
\item[(b)]
$\begin{pmatrix}
1 & 2 & 3 & 4 & 5 \\
4 & 2 & 5 & 1 & 3
\end{pmatrix}$
\item[(c)]
$\begin{pmatrix}
1 & 2 & 3 & 4 & 5 \\
3 & 5 & 1 & 4 & 2
\end{pmatrix}$
\item[(d)]
$\begin{pmatrix}
1 & 2 & 3 & 4 & 5 \\
1 & 4 & 3 & 2 & 5
\end{pmatrix}$
\end{enumerate}
\emph{Solution: }
\hfill $\square$
\vskip2ex

\vskip2ex

\textbf{Exercise 2(b, d, g, h, j, p).}
\vskip1ex
Compute each of the following.
\begin{enumerate}
\item[(b)]
$(12)(1253)$
\item[(d)]
$(1423)(34)(56)(1324)$
\item[(g)]
$(1254)^{-1} (123)(45) (1254)$
\item[(h)]
$(1254)^2(123)(45)$
\item[(j)]
$(1254)^{100}$
\item[(p)]
$[(1235)(467)]^{-1}$
\end{enumerate}
\emph{Solution: }
\hfill $\square$

\vskip2ex

\textbf{Exercise 3(b, d).}
\vskip1ex
Express the following permutations as products of transpositions and
identify them as even or odd.
\begin{enumerate}
 \item[(b)]
$(156)(234)$
 \item[(d)]
$(17254)(1423)(154632)$
\end{enumerate}
\emph{Solution: }
\hfill $\square$

\vskip2ex

\textbf{Exercise 4.}
\vskip1ex
Find $(a_1a_2 \ldots a_k)^{-1}$.
\vskip1ex
\emph{Solution: }
\hfill $\square$
\vskip2ex

\textbf{Exercise 6.}
Find all of the subgroups in $A_4$. What is the order of each subgroup?
\vskip1ex
\emph{Solution: }
\hfill $\square$
\vskip2ex

\textbf{Exercise 13.}
Let $\sigma = \sigma_1 \cdots \sigma_m \in S_n$ be the product of disjoint cycles.
Prove that the order of $\sigma$ is the least common multiple of the lengths of the cycles $\sigma_1, \ldots, \sigma_m$.
\vskip1ex
\emph{Proof: }
\hfill $\square$
\vskip2ex


\textbf{Exercise 22.}
If $\sigma$ can be expressed as an odd number of transpositions, show that any other product of transpositions equaling $\sigma$ must also be odd.
\vskip1ex
\emph{Proof: }
\hfill $\square$
\vskip2ex


\textbf{Exercise 23.}
If $\sigma$ is a cycle of odd length, prove that $\sigma^2$ is also a cycle.
\vskip1ex
\emph{Proof: }
\hfill $\square$
\vskip2ex


\textbf{Exercise 24.}
Show that a 3-cycle is an even permutation.
\vskip1ex
\emph{Proof: }
\hfill $\square$
\vskip2ex


\textbf{Exercise 27.}
Let $G$ be a group and define a map $\lambda_g : G \rightarrow G$ by $\lambda_g(a) = g a$. Prove that $\lambda_g$ is a permutation of $G$.
\vskip1ex
\emph{Proof: }
\hfill $\square$
\vskip2ex

\textbf{Exercise 34.}
If a permutation $\alpha$ is even, prove that $\alpha^{-1}$ is also even. Does a corresponding result hold if $\alpha$ is odd?
\vskip1ex
\emph{Proof: }
\hfill $\square$
\vskip2ex

\textbf{Exercise 35.}
If $\sigma \in A_n$ and $\tau  \in S_n$, show that $\tau^{-1}\sigma\tau \in A_n$.
\vskip1ex
\emph{Proof: }
\hfill $\square$

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