Mathematics of Sudoku

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Sudokus from group tables

As in the case of Latin squares the (addition- or) multiplication tables (Cayley tables) of finite groups can be used to construct Sudokus and related tables of numbers. Namely, one has to take subgroups and quotient groups into account:

Take for example ${\mathbb{Z}}_n\oplus {\mathbb{Z}}_n$ the group of pairs, adding each component separately modulo some $n$.

By omitting one of the components, we suddenly find ourselves in ${\mathbb{Z}}_n$ (and this mapping is obviously compatible with the respective additions, i.e. it is a group homomorphism).

One also says that the latter is a quotient group of the former, because some once different elements become equal in the new group.

However, it is also a subgroup, because we can simply fill the missing component with $0$ to get back to ${\mathbb{Z}}_n\oplus {\mathbb{Z}}_n$.