Cosets

The product group $G=({\mathbf{Z}}_m\times {\mathbf{Z}}_n,+)$ contains subgroups $H=⟨S⟩$ 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 $({\mathbf{Z}}_8\times {\mathbf{Z}}_{12},+)$. Click one or more cells to add elements to $S$; the elements of $H=⟨S⟩$ 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.

$S=\{\}$.

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	0	1	2	3	4	5	6	7

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

$m=$ $n=$