The existence of Hall subgroups can be proved by induction on the order of G, using the fact that every finite solvable group has a normal elementary abelian subgroup. More precisely, fix a minimal normal subgroup A, which is either a or a as G is . By induction there is a subgroup H of G containing A such that H/A is a Hall of G/A. If A is a then H is a Hall of G. On the other hand, if A is a , then by the Schur–Zassenhaus theorem A has a complement in H, which is a Hall of G.
