Exercise 1.3.6 (Cycle decomposition of the elements of order 4 in $S_4$)

Question

Write out the cycle decomposition of each element of order $4$ in $S_4$.

Richard Ganaye · Accepted Answer

\begin{proof} We keep only the elements of order 4 in the array of Exercise 4(b)
$$
\begin{array}{c|c}
\sigma & |\sigma|\
\hline
 (1 \ 2 \ 3 \ 4)&4\
 (1 \ 4 \ 3 \ 2)&4\
 (1 \ 3 \ 4 \ 2)&4\
 (1 \ 3 \ 2 \ 4)&4\
 (1 \ 4 \ 2 \ 3)&4\
 (1 \ 2 \ 4 \ 3)&4\
 \end{array}
$$
So all the elements of order $4$ in $S_4$ are cycles of length $4$.
\end{proof}

Exercise 1.3.6 (Cycle decomposition of the elements of order 4 in $S_4$)

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Exercise 1.3.6 (Cycle decomposition of the elements of order 4 in $S_4$)

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