Direct product for minimal normal subgroup

Question

Given a finite group $G$, $N$ is a minimal normal subgroup and $S_i$ are simple subgroups of $G$ for $i=1, \cdots , r$.

Next, the problem said "suppose $N = S_1\times \cdots \times S_r$"

First does this notation always mean the internal direct product in group theory? Since it has strong implications such as

1. each $S_i$ are normal in $N$
2. $S_i \cap S_j = id$ for $i\neq j$
3. $S_i$ and $S_j$ commute for $i\neq j$.

Second, the problem asked to show that for each $g\in G$, $gS_1g^{-1} = S_i$ for some $i$.

The solution just said it is true because the minimal normal subgroup of $N$ are of the form $id \times \cdots \times S_i \times \cdots id$, why does this imply the result?

Thank you.