Page 105 says - A careful look at Figure 6.9 reveals that the cosets of $\left< \, (0,1) \,\right>$ partition $C_3 \times C_3$.
How is this true? The picture shows $gH = left picture = (1,0)\left< \, (0,1) \,\right>= \{(1, 0), (1, 1), (1, 2)\} \\ = right picture = \left< \, (0,1) \, \right>(1,0) = Hg $.
But this is not all of $C_3 \times C_3$ therefore not a partition?
The book hasn't introduced quotient groups or normal subgroups still. This is from Nathan Carter page 104 Visual Group Theory.
The partition consists of $G/\langle (0,1) \rangle = \{(0,0)\langle (0,1) \rangle,\ (1,0)\langle (0,1) \rangle,\ (2,0)\langle (0,1) \rangle\}$. He never says that $gH$ is a partition of $G$; it is an element of the partiton, that, in this case, is $\{g^0H,\ gH, \ g^2H\}$.