Minimal generating set for product of groups

Question

Let $S_1$ be a minimal generating set for a group $G_1$, $S_2$ be a minimal generating set for a group $G_2$. $T_j:G_j\to G_1\times G_2, j=1,2 $. Then can we say $T(S_1) \cup T(S_2)$ is a "minimal" generating set for $G_1\times G_2$?

Answer 1

It is a generating set, which we can see as follows. Given any element $g$ of $G_1 \times G_2$, $g$ has the form $(g_1, g_2)$. We can express $(g_1, 1)$ as a combination of elements from $T_1(S_1)$, and similarly $(1, g_2)$ as a combination of elements from $T_2(S_2)$. Then the product of those two combinations gives $g$ as a combination of elements from $T_1(S_1) \cup T_2(S_2)$.

However, it is not always a minimal generating set, as @RobertChamberlain points out. For $G_1 = \Bbb{Z}/2$ and $G_2 = \Bbb{Z}/3$, a minimal generating set for each of the $G_i$ has a single generator, call it $\gamma_i$. The combined generating set is $\{(\gamma_1, 1), (1, \gamma_2)\}$, but $G_1 \times G_2$ is cyclic, so a minimal generating set for $G_1 \times G_2$ has only one element. In fact $(\gamma_1, \gamma_2)$ generates $G_1 \times G_2$.