I have a quick question. Are all the subgroups of $S_n$ symmetric as well?
2026-04-01 22:54:25.1775084065
Are the subgroups of $S_n$ symmetric?
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If you mean, "is every subgroup of a symmetric group isomorphic to a symmetric group?", the answer is no.
For example, look at the subgroup generated by a $k$-cycle, with $k>2$ inside of a symmetric group of order greater than $2$. This will be a cyclic group, and no symmetric group $S_n$ is cyclic for $n>2$.