Nine lemma

In mathematics, the nine lemma (or 3×3 lemma) is a statement about commutative diagrams and exact sequences valid in the category of groups and any abelian category.

Similarly, because the diagram is symmetric about its diagonal, rows and columns may be interchanged in the above as well.

The nine lemma can be proved by direct diagram chasing, or by applying the snake lemma (to the two bottom rows in the first case, and to the two top rows in the second case).

Linderholm (p. 201) offers a satirical view of the nine lemma: Draw a noughts-and-crosses board... Do not fill it in with noughts and crosses...

Make some noughts, but not in the squares; put them at both ends of the horizontal and vertical lines.