I am currently working on the book The Symmetric Group by Bruce Sagan.
The following passage comes before introducing Specht Modules:
Suppose that the tableau $t$ has rows $R_1, R_2, ..., R_l$ and columns $C_1, C_2, ..., C_k$. Then $$R_t = S_{R_1} \times S_{R_2} \times ... \times S_{R_l}$$ and $$C_t = S_{C_1} \times S_{C_2} \times ... \times S_{C_k}$$ are the row-stabiliser and the column-stabiliser of $t$, respectively.
We define $\kappa_t = \sum \limits_{\pi \in C_t} \text{sgn}(\pi) \pi$.
Note that $\kappa_t$ factors as $$\kappa_t =\kappa_{C_1}\kappa_{C_2}...\kappa_{C_k}.$$
Unfortunately I cannot see a way how to prove that this is true. I think that the right hand side represents a product of group subsets.
So, using the definitions, I am wondering, why $$\sum \limits_{\pi \in C_t} \text{sgn}(\pi) \pi = \sum \limits_{\pi \in C_1} \text{sgn}(\pi) \pi\sum \limits_{\pi \in C_2} \text{sgn}(\pi) \pi~...\sum \limits_{\pi \in C_k} \text{sgn}(\pi) \pi.$$
Do you know how to prove this?
Thank you very much for your help!