If $G$ is a connected solvable Lie-group, then $[G,G]$ is nilpotent. The corresponding statement for Lie algebras follows from Lie's theorem, and it then follows from connected Lie groups by exponentiation.
Is the statement also true for finite groups? I can't find a counterexample, but I didn't try that hard.
Motivation: I'm prepping to teach a group theory and Galois theory course, so I'm brainstorming interesting questions about finite groups. Whenever I come up with one I can't answer, I plan to toss it up here or at MO.