I love the proof of Wilson's theorem through consideration of the multiplicative group of integers modulo-p. The solution is very slick and it wasn't until I saw this early in an undergraduate course that I truly understood the power of group theory and its wide range of applicability. It also has a nice bit of history to it: After neither Wilson nor his mentor Warring was able to give a proof of the assertion, Gauss remarked in Disquisitiones Arithmeticae: "And Waring confessed that the proof seemed the more difficult, since one cannot imagine any notation to express a prime number. – In our opinion, however, such truths should be extracted from notions rather than from notations".

I love the proof of Wilson's theorem through consideration of the multiplicative group of integers modulo-p. The solution is very slick and it wasn't until I saw this early in an undergraduate course that I truly understood the power of group theory and its wide range of applicability. It also has a nice bit of history to it: After neither Wilson nor his mentor Waring was able to give a proof of the assertion, Gauss remarked in Disquisitiones Arithmeticae: "And Waring confessed that the proof seemed the more difficult, since one cannot imagine any notation to express a prime number. – In our opinion, however, such truths should be extracted from notions rather than from notations".