Cartesian Product of Permutations?

[Parmenides]

Suppose I was asked if $G \cong H \times G/H$. At first I considered a familiar group, $G = S_3$ with its subgroup $H = A_3$. I know that the quotient group is the cosets of $H$, but then I realized that I have no idea how to interpret a Cartesian product of any type of set with elements that aren't just numbers. An ordered pair of permutations doesn't make sense (this is not a homework question). I'd be grateful for some clarity.

 

[Office_Shredder]

If G₁ and G₂ are groups, then
$$G_1 \times G_2 = \{ (g_1,g_2)\ :\ g_1 \in G_1,\ g_2\in G_2 \}$$
with the multiplication
$$(g_1,g_2)\cdot (h_1,h_2) = (g_1 h_1, g_2 h_2)$$.

So if you have G = S₃, and H = A₃, G/H is isomorphic to the two element group {1,-1} (where each permutation gets mapped to its parity), and a general element of A₃ x (S₃/A₃) is $(\sigma, \pm 1 )$ where sigma here is any even permutation.

For example,
$$\left( (1 2 3 ),-1 \right) \cdot \left( (1 2 3), 1 \right) = \left( (1 3 2), -1 \right)$$
is a multiplication inside of this group.