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.