Two famous problems about finite groups were solved by showing that the corresponding infinitesimal Lie rings are nilpotent (of a certain class).

We first introduce the two problems, and we then sketch the ideas that go into the proofs (e.g.: the linearity of groups, fix-points of automorphisms, the classification of the finite simple groups, and the factorisation of circulant determinants).

This is a colloquium-style talk intended for a general audience.