Cosets

The product group $G = (\mathbf{Z}_{m} \times \mathbf{Z}_{n}, +)$ contains subgroups $H = \langle S \rangle$ generated by zero or more elements. The left cosets and right cosets of $H$ coincide because $G$ is Abelian.

The cells of the array correspond to elements of . Click one or more cells to add elements to $S$; the elements of $H = \langle S \rangle$ are highlighted.

At any stage you may “freeze” the subgroup $H$ under consideration and investigate its cosets. Once $H$ is frozen, clicking a cell $(a, b)$ highlights elements of the coset $(a, b) + H$ in a different color.

Select $m$ and $n$, then click the button to generate a new group. (Maximum size is .)

$m = $ $n = $