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Does the permutation group $S_8$ contain elements of order $14$?My answer: If $\sigma =\alpha \beta$
where $\alpha$ and $\beta$ are disjoint cycles, then
$|\sigma|=lcm(|\alpha|, |\beta|)$ .
Therefore the only possible disjoint cycle decompositions for a permutation $\sigma \in S_8$ with $|\sigma| =14$ is $(7,2)$. Since $7+2\neq 8$ so there is no element of order 14 in $S_8$.
Is my answer right?
where $\alpha$ and $\beta$ are disjoint cycles, then
$|\sigma|=lcm(|\alpha|, |\beta|)$ .
Therefore the only possible disjoint cycle decompositions for a permutation $\sigma \in S_8$ with $|\sigma| =14$ is $(7,2)$. Since $7+2\neq 8$ so there is no element of order 14 in $S_8$.
Is my answer right?
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