# Young subgroup

Let $\{1,2,\ldots,n\} = \cup_{i=1}^k \alpha_i$ be a partition of $\{1,2,\ldots,n\}$ into $k$ disjoint subsets. Then the corresponding Young subgroup of $S_n$, the symmetric group on $n$ letters, is the subgroup $$S_{\alpha_1} \times \cdots \times S_{\alpha_k} \,,$$ where $S_{\alpha_i} = \{ \sigma \in S_n : \sigma(j) = j \ \text{for all}\ j \not\in \alpha_i \}$. Sometimes only the particular cases $$S_{\alpha_1} \times \cdots \times S_{\alpha_k}$$ are meant where $\alpha_i = \{\lambda_{i-1} + 1,\ldots, \lambda_i\}$, where $\lambda_0 = 0$ and $\lambda = (\lambda_1,\ldots,\lambda_k)$ is a partition of the natural number $n$, i.e. $\lambda_1 \ge \lambda_2 \ge \cdots \ge \lambda_k$, $\sum \lambda_i = n$.