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The things you're talking about would not be accessible to the youngsters I deal with, so I'm looking for something they wouldn't be able to show, but which becomes obvious with just basic group theory.
It's not quite that level, but Fermat's little theorem of course follows from Lagrange's theorem. I'll bet you could do an amazing elementary video that was Euclid's algorithm, Fermat's little theorem, Lagrange's theorem, and Pratt's theorem (!), all in about 15 minutes!
If you haven't seen it, Pratt's theorem is an amazing result about primes: https://en.wikipedia.org/wiki/Primality_certificate … Easy to prove once you know Fermat's little theorem.
It's kind of remarkable: you can start from the question "What does it take to prove a number is prime?", and quickly end up being almost forced to reinvent Lagrange's theorem (and groups and cosets) and Euclid's algorithm and Fermat's little theorem.
This is very close to what I currently do, but it suffers the problem that it all seems to be about numbers. You and I know that it isn't, but when you're trying to convince people that "real maths" isn't about numbers, it's a but awkward to do Fermat's Little Theorem.
You show Pratt's theorem?
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