Suppose $ G_{1}, \dots, G_{d}$ are permutation groups.
Given $ G_{i}$ some finite groups for $ i=1,...,d$, we define the direct product group $ G_{1}\times ... \times G_{d} $ as the Cartesian product, $ G_{1} \times ... \times G_{d} $ $^{1}$ where multiplication is defined coordinate-wise $ (g_{1},...,g_{d}) \cdot (g_{1}^{\prime},...,g_{d}^{\prime}) = (g_{1} \cdot g^{\prime}_{1},...,g_{d} \cdot g_{d}^{\prime})$.
My question is:
Should the generators of $ G_{1}, ... , G_{d}$ be disjoint? Can you give me any reference please?
I think they must be disjoint since the underlying set must is partitioned .
$^1$ That is, the ordered d-tuples $ (g_{1},...,g_{d}) $, where $ g_{1}\in G_{1} $ ,..., $ g_{d}\in G_{d}$.