Minimal normal subgroup
= Minimal normal subgroup
A minimal normal subgroup is a nontrivial <normal subgroup> containing no smaller nontrivial normal subgroup of the ambient group. It need not be a minimal nontrivial subgroup. In a finite <soluble group>, it is elementary abelian: its derived subgroup must be trivial, a characteristic Sylow subgroup selects one prime, and the characteristic subgroup of elements killed by that prime gives exponent $p$.