Pictorial representations of the isomorphism theorems

I'm looking for a graphic that shows the isomorphism theorems illustrated using a finite vector space. (that is a finite dimensional vector space over a finite field.) Something that shows nicely how the cosets "collapse" in to single elements or how cosets at different "levels" are matched up to form the isomorphisms.

I have tried searching but I keep finding arrow diagrams.

Alternately, suggestions for good examples that don't take too long to draw would be appreciated.

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@a little don: A simple enough picture can be done with $\mathbb{R}^2$ and $H=\{(ka,kb): x\in\mathbb{R}\}$. The cosets are the lines of slope $b/a$, and if you pick your favorite line through the original that is not of that slope, the cosets "colapse" to their projection onto that line. E.g., $H=\{(x,0):h\in \mathbb{R}\}$, and project onto the $Y$-axis to get $\mathbb{R}^2/H\cong \mathbb{R}$. – Arturo Magidin Nov 29 '10 at 20:52

2 Answers

Over 9000 hours in MSPaint and I now have the graphic Arturo constructed in his comment. Strikingly, it's also what I've always pictured in my head when I visualize cosets and quotients:

Each colored line is a copy of $H$ translated, and they are each individually the coset of the bold color dot where it intersects the gray line; you can see quite vividly how each colored line "collapses" into a dot on the gray one. If you haven't figure it out already, the gray line is $\mathbb{R}^2/H$.

As for the full isomorphism theorems, actually illustrating them rather than simply diagramming them should prove to be a difficult but rather rewarding project. I hope to see some talented artist take up the challenge in the future.

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This is for $\mathbb{R}^n$, though it applies to $\mathbb{F}_p^n$ just as well since it is more of a metaphorical picture (e.g., the "orthogonality" of the subspaces).

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