This is something that Lambek used for diagram chasing in his book on Rings and Modules. I have discovered a very simple proof of a slight generalization. The lemma states (as least Jim stated it this way) that if G is a submodule of A x B and C = image of G --> A, C' the image of G --> B, D the kernel of G --> B and D' the kernel of G --> A, then C/D is isomorphic to C'/D'. Forget that G is a subgroup of A x B and just suppose you have two exact sequences 0 --> D' --> G --> C --> 0 and 0 --> D --> G --> C' --> 0, then the composites D --> G --> C and D' --> G --> C' have the same cokernel. The dual claim is that they have the same kernel. But thinking of D and D' are submodules of G, then the kernels are quite obviously D \cap D' and the original conclusion follows by duality. However, the same thing is true for groups (both D and D', being kernels, are normal in G and hence in C, C'). I see no such simple duality argument there. Michael [For admin and other information see: http://www.mta.ca/~cat-dist/ ]