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QUESTION


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\item[(a)]
Compute the following products of the permutations
$\sigma=(1576)(234),\ \tau=(132)(4675)$ expressing your answers in
disjoint cycle notation:

\begin{description}

\item[(i)]
$\sigma\tau.$

\item[(ii)]
$\tau\sigma.$

\item[(iii)]
$\sigma^2\tau^{-1}.$

\end{description}

\item[(b)]
Express the permutation
$\sigma=\left(\begin{array}{ccccccc}1&2&3&4&5&6&7\\4&6&1&3&7&5&2\end{array}\right)$
in disjoint cycle notation and as a product of transpositions. Fin
the order and sign of $\sigma$ and calculate $\sigma^{2001}$.

\item[(c)]
List all the possible cycle structures for elements of $S_7$ and
use this to find all the possible orders for elements of $S_7$.

\item[(d)]
Find all the possible cycle structures corresponding to elements
of order 6 in $S_9$, and compute the number of elements of $S_9$
corresponding to each cycle structure.

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ANSWER


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\item[(a)]

\begin{description}

\item[(i)]
$\sigma\tau=(14)(25)$

\item[(ii)]
$\tau\sigma=(14)(36)$

\item[(iii)]
$\sigma^2\tau^{-1}=(14637)$

\end{description}

\item[(b)]
$\sigma=(143)(2657)=(14)(43)(26)(65)(57)$ has order 12 and sign
$-1$. $2001=166.12+8$ so $\sigma^{2001}=\sigma^9$. Now $\sigma^9$
generates the same subgroup of $\left<\sigma\right>$ as does
$\sigma^3$, hence it has order 4.

\item[(c)]

\begin{tabular}{c|c}
Cycle structure&order\\ \hline $\left[7\right]$&7\\
$\left[6\right]$&6\\ $\left[5,2\right]$&10\\ $\left[5\right]$&5\\
$\left[4,3\right]$&12\\ $\left[4,2\right]$&4\\
$\left[4\right]$&4\\ $\left[3,3\right]$&3\\
$\left[3,2,2\right]$&6\\ $\left[3,2\right]$&6\\
$\left[3\right]$&6\\ $\left[2,2,2\right]$&2\\
$\left[2,2\right]$&2\\ $\left[2\right]$&2\\ $\left[1\right]$&1
\end{tabular}

\item[(d)]
The possible cycle structures are
$\left[3,3,2\right],\left[3,2,2\right],\left[3,2\right],\left[6,3\right],\left[6,2\right],\left[6\right]$.
There are respectively 9.8.7.6.5.4.3.2/(3.3.2.2),
9.8.7.6.5.4.3/(3.2.2.2), 9.8.7.6.5/(3.2), 9.8.7.6.5.4.3.2.1/(6.3),
9.8.7.6.5.4.3.2/(6.2) and 9.8.7.6.5.4/6 elements with these
structures.

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