For any group a minimal normal subgroup is a normal subgroup of such that the only normal subgroups of which are contained in are and 1. Prove that every minimal normal subgroup of a finite solvable group is an elementary abelian -group for some prime .
Let be a finite solvable group and let be a minimal normal subgroup.
If , then the conclusion is trivial. Suppose .
Since is solvable, is solvable. Since , is proper; now is characteristic in , hence normal in . Since is minimal, we have ; thus is abelian.
Recall from this previous exercise that is characteristic in , hence normal in , for all primes . Thus for all primes . By Cauchy, some element of has prime order, so that for some prime ; in particular . Thus is an elementary abelian -group.