Order of elements in a symmetric group
Let
- Give an example of two non-conjugate elements of
that have the same order. - If
has maximal order, what is the order of ? - Does the element
that you found in part (2) lie in ? Fully justify your answer. - Determine whether the set
is a single conjugacy class in , where is the element you found in part (2).
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Let $S_7$ denote the symmetric group.
\begin{enumerate}[label=\alph*)]
\item Give an example of two non-conjugate elements of $S_7$ that have the same order.
\item If $g\in S_7$ has maximal order, what is the order of $g$?
\item Does the element $g$ that you found in part (b) lie in $A_7$? Fully justify your answer.
\item Determine whether the set $\{h\in S_7\mid |h|=|g|\}$ is a single conjugacy class in $S_7$, where $g$ is the element you found in part (b).
\end{enumerate}