Show $S_n$ contains a subgroup isomorphic to direct product of $S_{n_1}, ..., S_{n_k}$...

Question

Let $n=n_1+n_2+...+n_k$ be a partition of the positive integer $n$ , then how do I show that the symmetric group $S_n$ contains a subgroup which is isomorphic to direct product of $S_{n_1},S_{n_2} , ..., S_{n_k}$ ?

Answer 1

Consider the subgroup generated by the following $k$ disjoint $n_i$ cycles:

$$\Big(1\cdots n_1\Big),\;\Big((n_1+1)\cdots (n_1+n_2)\Big),\;\cdots\;,\;\Big((n_1+\cdots+n_{k-1}+1)\cdots (n_1+\cdots+n_k)\Big)$$ Together with:

$$\Big(12\Big),\;\Big((n_1+1)(n_1+2)\Big),\;\cdots\;,\;\Big((n_1+\cdots+n_{k-1}+1)(n_1+\cdots+n_{k-1}+2)\Big)$$

Now since $(123\cdots m)$ together with $(12)$ generate $S_m$, we will find that in this new subgroup we can individually permute the first $n_1$ elements in any way we want, the same holds for the $n_2$ elements after that etc... But we can not exchange any elements between two "blocks" of $n_i$ elements. Now convince yourself that this is isomorphic to $S_{n_1}\times\cdots\times S_{n_k}$.

Answer 2

Split the elements into disjoint sets such that their cardinalities match the partition. Now, consider the set of those permutations that permutes only elements in the same set.