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I'm not a group theorist, nor a mathematician, so I'm the wrong person to ask. I do find the Solovay-Kitaev thm wonderful: informally, if you take products of elements in (many) compact Lie groups, they fill in the group exponentially quickly, & surprisingly near to uniformly
A picture is better to explain what this means. And probably best explained concretely using, e.g., rotations and a few similar examples.
I don't quite understand it well enough to make the attempt, but I'll bet there's a mindblowing 5-minute explanation of what Gromov's theorem on groups of polynomial growth says.
Hmm. I'll have to think about that. I certainly don't know enough to be able to see immediately how or why that's interesting, so I'd have to dig into it. One result I did use to impress some 16 year olds was that in a group of even order there is an element a with a^2 = 1.
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