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QUESTION
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\item[(a)]
The element $\sigma$ is an element of the finite permutation group
$S_n$. Explain the relationship between the cycle structure if
$\sigma$ and its order, and use this to find the smallest positive
integer $n$ such that $S_n$ contains an element of order 12.
List the possible cycle structures for elements of order 14 in
$S_9$ and use this to find the number of such elements. (You are
NOT required to list them all.)
\item[(b)]
Express the permutation
$\sigma=\left(\begin{array}{ccccccccc}1&2&3&4&5&6&7&8&9\\
2&4&7&6&3&5&1&9&8\end{array}\right)$ in disjoint cycle notation
and as a product of transpositions. Fine the order and sign of
$\sigma$ and calculate the order of $\sigma^{2000}$.
\item[(c)]
Say what it means for a permutation in the symmetric group $S_n$
to be even, and show that a permutation is even if and only if it
can be written as a product of 3-cycles.
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ANSWER
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\item[(a)]
The order of $\sigma$ is the least common multiple of the lengths
of its cycles. Ignoring 1-cycles the possible cycle structures for
an element of order 12 are $\left[12\right],\ \left[3,4\right]$
and the smallest $n$ such that $S_n$ contains an element of order
12 is 7.
$\left[2,7\right]$ is the only possible cycle structure. There are
$\frac{9.8}{2}$ possible transpositions and $6!$ different 7-
cycles so $36.720=25,920$ different elements of order 14.
\item[(b)]
$\sigma=$(1 2 4 6 5 3 7)(8 9)=(1 2)(2 4)(4 6)(6 5)(5 3)(3 7)(8 9)
which has order 14 and sign $-1$.
$2000=(14.142)+12$ so $\sigma^{2000}=\sigma^{12}$ and
$\sigma^{12}$ has order 7.
\item[(c)]
A permutation is even $\Leftrightarrow$ it can be written as a
product of an even number of transpositions.
Any 3-cycle $(x\ y\ z)$ can be written as $(x\ y\ z)=(x\ y)(y\ z)$
so any product of 3-cycles is even.
Any pair of transpositions can be written as a 3-cycle and any
poair $x\ y)(u\ v)$ can be written as the product $(x\ y)(y\
u)(y\ u)(u\ v)$ so as the product $(x\ y\ u)(y\ u\ v)$.
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