Exercise 1.3.18 (Order of the elements of $S_5$)

Question

Find all numbers $n$ such that $S_5$ contains an element of order $n$. [Use Exercise 15.]

Richard Ganaye · Accepted Answer

\begin{proof} 
Let $\sigma$ be any element of $S_5$.

By Exercise 15, the order $\sigma$ equals the least common multiple of the lengths of the cycles in its decomposition.

The ways to decompose $5$ in some of integers are 
$$(1, 1, 1, 1, 1),\ (2, 1, 1, 1),\ (2, 2, 1),\ (3, 1, 1),\ (3, 2),\ (4, 1),\ (5).$$
Each way corresponds to a type of cycle decomposition, so the cycle decomposition of $\sigma$ is one of the following forms
\begin{center}
\begin{tabular}{|c|c|}
\hline
$\sigma$ & $|\sigma|$     \
\hline
$e$ & $1$\
$(a\ b)$ & $2$    \
 $(a\ b)(c\ d)$&  $2$   \
 $(a\ b\ c)$ & $3$\
 $ (a\ b\ c)(e\ f)$ & $6$   \
 $(a\ b \ c \ d)$ & $4$\
 $(a\ b \ c \ d \ e)$& $5$\
\hline
\end{tabular}
\end{center}
So the orders of the permutations of $S_5$ are $1,2, 3,4,5,6$.
\end{proof}

Verification with Sagemath:
\begin{verbatim}
sage: G = SymmetricGroup(5); G
Symmetric group of order 5! as a permutation group
sage: s = set()
sage: for sigma in G:
....:     s.add(sigma.cycle_type())
....: s
....: 
{[1, 1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1], [3, 1, 1], [3, 2], [4, 1], [5]}
sage: s = set()
sage: for sigma in G:
....:     s.add(lcm(sigma.cycle_type()))
....: s
....: 
{1, 2, 3, 4, 5, 6}
\end{verbatim}

Exercise 1.3.18 (Order of the elements of $S_5$)

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$σ$	$\| σ \|$

$e$	$1$
$(a b)$	$2$
$(a b) (c d)$	$2$
$(a b c)$	$3$
$(a b c) (e f)$	$6$
$(a b c d)$	$4$
$(a b c d e)$	$5$

Exercise 1.3.18 (Order of the elements of $S_5$)

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