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 $p$.