# 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: $\S 79 \delta$