Subgroups of Symmetric Group Isomorphic to Product of Subgroups
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Theorem
Let $S_n$ denote the symmetric group on $n$ letters.
Let $k \in \closedint 1 n$.
Then there are $\dbinom n k$ subgroups of $S_n$ which are isomorphic to $S_k \times S_{n - k}$, where $\dbinom n k$ denotes the binomial coefficient.
All of these $\dbinom n k$ subgroups are conjugate.
Proof
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Sources
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: The Symmetric Groups: $\S 79 \delta$