- Thread starter
- #1

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?

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?

Last edited: