Cyclic normal Sylow subgroup for least prime divisor is central

Statement
Suppose $$p$$ is the least prime divisor of the order of a finite group $$G$$. Suppose $$S$$ is a $$p$$-Sylow subgroup that is also a cyclic normal subgroup. In other words, $$S$$ is a normal Sylow subgroup that is cyclic as a group. Then, $$S$$ is a central subgroup of $$G$$.

Other facts about least prime divisor

 * Normal of least prime order implies central
 * Subgroup of least prime index is normal

Applications

 * Cyclic Sylow subgroup for least prime divisor has normal complement