Normal of order equal to least prime divisor of group order implies central

Statement
In a finite group, a normal subgroup whose order is the least prime divisor of the order of the group, must be a central subgroup (viz, it is contianed inside the center).

Similar facts

 * Normal of order two implies central
 * Cyclic normal Sylow subgroup for least prime divisor is central, used to show that cyclic Sylow subgroup for least prime divisor has normal complement
 * Minimal normal implies central in nilpotent, leading to minimal normal implies contained in Omega-1 of center for nilpotent p-group or equivalently socle equals Omega-1 of center in nilpotent p-group

Dual facts

 * Subgroup of index equal to least prime divisor of group order is normal
 * Index two implies normal

Other normal-to-central facts

 * Totally disconnected and normal in connected implies central
 * Cartan-Brauer-Hua theorem

Proof
Take any element inside the normal subgroup. The size of the conjugacy class of that element is strictly less than the least prime divisor (because the identity element is in a different conjugacy class from other elements). Since the conjugacy class of the element is an orbit of the element under the group action under conjugation, the size of the conjugacy class divides the order of the group. The only divisor of the order of the group, which is less than the least prime divisor, is 1. Thus every conjugacy class has size 1, and the subgroup is thus contained in the center.