  We prove that every finite, commutative automorphic loop is solvable. We also
prove that every finite, automorphic 2-loop is solvable. The main idea of the
proof is to associate a simple Lie algebra of characteristic 2 to a
hypothetical finite simple commutative automorphic loop. The "crust of a thin
sandwich" theorem of Zel'manov and Kostrikin leads to a contradiction.

Group Theory